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arXiv:1206.4431v2 [math.GR] 25 Jul 2012
Cyclic rewriting and conjugacy problems
Volker Diekert and Andrew J. Duncan and Alexei Myasnikov
July 26, 2012
Abstract
Cyclic words are equivalence classes of cyclic permutations of ordi-
nary words. When a group is given by a rewriting relation, a rewrit-
ing system on cyclic words is induced, which is used to construct
algorithms to find minimal length elements of conjugacy classes in
the group. These techniques are applied to the universal groups of
Stallings pregroups and in particular to free products with amalga-
mation, HNN-extensions and virtually free groups, to yield simple
and intuitive algorithms and proofs of conjugacy criteria.
Contents
1 Introduction 2
2 Preliminaries 4
2.1 Transposition, conjugacy and involution . . . . . . . . . . . . . 4
2.2 Rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Thue systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Cyclic words and cyclic rewriting . . . . . . . . . . . . . . . . . 7
2.5 From confluence to cyclic confluence . . . . . . . . . . . . . . . 9
2.6 From strong to cyclic confluence in groups . . . . . . . . . . . . 12
2.7 A Knuth-Bendix-like procedure on cyclic words . . . . . . . . 15
2.8 Strongly confluent Thue systems . . . . . . . . . . . . . . . . . 17
2.9 Cyclic geodesically perfect systems . . . . . . . . . . . . . . . . 18
3 Stallings’ pregroups and their universal groups 20
3.1 Amalgamated products and HNN-extensions . . . . . . . . . . 23
3.2 Fundamental groups of graph of groups . . . . . . . . . . . . . 24
4 Conjugacy in universal groups 24
1
5 The conjugacy problem in amalgamated products and HNN-
extensions 31
5.1 Conjugacy in amalgamated products . . . . . . . . . . . . . . . 31
5.2 Conjugacy in HNN-extensions . . . . . . . . . . . . . . . . . . . 32
6 The conjugacy problem in virtually free groups 34
1 Introduction
Rewriting systems are used in the theory of groups and monoids to specify
presentations together with conditions under which certain algorithmic prob-
lems may be solved. Typically, presentations given by convergent rewriting
systems are sought as these give rise to algorithms for the word and geodesics
problems. Recently, less stringent conditions on rewriting systems which still
allow the word problem and/or the geodesics problem to be decided, have also
been investigated: for example geodesic or geodesically perfect systems[10, 7].
In contrast to the classical case, geodesically perfect systems are confluent,
but not necessarily convergent or finite, and are designed to seek geodesics in
a group, rather than normal forms of elements. In any case, all these systems
depend on rewriting of strings of letters, or words, from the free monoid on
the generating set of a group or monoid.
In this paper we consider applications of rewriting systems to the conju-
gacy problem in groups. To this end we apply rewriting to cyclic words rather
than ordinary words. Cyclic words can be viewed as sets of all cyclic per-
mutations of standard words or, equivalently, as graphs, which are directed
labelled cycles. This allows us to construct algorithms for finding represen-
tatives of minimal length in the conjugacy classes of elements in groups.
We describe analogues of Knuth-Bendix completion for rewriting systems
on cyclic words and consider how to realise these procedures in particular sit-
uations. Our approach to the completion processes on cyclic words is rather
different from the one developed by Chouraqui in [6], where cyclic rewriting
systems are used to construct algorithmic solutions to the conjugacy and
transposition problems in monoids, under suitable conditions. One signifi-
cant difference is that we introduce certain new rewriting rules, which are
specific to the cyclic rewriting. These rules are absolutely essential but are
not “induced” by standard string rewriting. Furthermore, in [6] the rewrit-
ing systems considered are all finite, whereas here we allow infinite systems.
As in the case of geodesically perfect string rewrite systems we do not re-
quire our systems to be convergent. This allows us to construct confluent
cyclic rewriting systems which are particularly suitable for working with the
2
conjugacy problem in groups.
We apply these techniques to the conjugacy problem in universal groups
of Stallings pregroups and fundamental groups of graphs of groups. As a
warm-up we give short intuitive proofs of the conjugacy criteria of free prod-
ucts with amalgamation [21] and in HNN-extensions (Collin’s Lemma) [20].
Moreover we are able to describe a linear time algorithm for the conjugacy
problem in finitely generated virtually free groups. (Epstein and Holt [8]
have constructed a linear time algorithm for the conjugacy problem in arbi-
trary hyperbolic groups. However, for this special case we give a very simple
construction based on the underlying finite rewriting system.)
Canonical examples of pregroups and their universal groups arise from
free products with amalgamation, HNN-extensions and, more generally, fun-
damental groups of graphs of groups. The conjugacy problem may behave
badly with respect to these constructions: for example in [18] an HNN ex-
tension G=HNN(H, t;t−1at =b)is constructed, where the base group H
has solvable conjugacy problem and the elements aand bof Hare infinite
cyclic, but Ghas unsolvable conjugacy problem. Several authors have stud-
ied conditions under which amalgams, HNN-extensions and graphs of groups
do have solvable conjugacy problem, see for example [12, 13, 14, 19] and
the references therein. Our results show that the obstruction to deciding
the conjugacy problem in such groups arises only from the determination
of conjugacy of elements of length one, with respect to the corresponding
pregroup. Thus, if the conjugacy problem in the group is undecidable our
systems do not provide a computable rewriting, but they do indicate where
the difficulties are. This also gives a different view-point on the results of pa-
pers [4, 3, 5, 9] where efficient generic algorithms for the conjugacy problem
in free products with amalgamation and HNN-extensions where constructed.
These algorithms are fast correct partial algorithms that give the answer on
most (”generic”) inputs, and do not give an answer only on a negligble set
of inputs.
The structure of the paper is as follows. In Section 2 we outline the
transposition and conjugacy problems for monoids and groups and give a
brief introduction to string rewriting systems. Section 2.4 contains the defi-
nitions of cyclic words, cyclic rewriting systems and the appropriate notions
of geodesic and geodesically perfect cyclic systems, needed later in the paper.
In Section 2.5 we consider how a semi-Thue system may be “completed” to
give a larger, semi-Thue, system which is confluent on cyclic words. This is
possible under a weak termination condition, but the price is that, in general,
length increasing rules may be introduced. This leads in 2.7, 2.8 and 2.9 to
consideration of analogues of Knuth-Bendix completion processes in which
we add context sensitive rules, that rewrite transposed words directly to each
3
other; when they are of some globally bounded length.
In Section 3 we describe Stallings pregroups, their universal groups and
the rewriting systems to which they are naturally associated. Section 4 con-
tains the main results on conjugacy in the universal groups of pregroups,
namely Theorem 4.4, Corollary 4.5 and Theorem 4.6. These results are ap-
plied to free products with amalgamation, HNN-extensions and virtually free
groups in Sections 5 and 6.
2 Preliminaries
2.1 Transposition, conjugacy and involution
Let Mbe a monoid and f , g ∈M. Then fand gare said to be transpose,
if there exist elements r, s ∈Msuch that f=rs and g=sr. We write f∼g
to denote transposition. The elements fand gare called conjugate, if there
exists an element z∈Msuch that fz =zg.
In general these definitions describe different relations. Indeed, conjugacy
is transitive, but not necessarily symmetric, while the transposition relation is
reflexive and symmetric, but not in general transitive. All transpose elements
are conjugate. If the monoid Mis a group, then conjugacy is an equivalence
relation and fand gare conjugate if and only if there exists an element
z∈Msuch that f=zgz−1.
Throughout Γ denotes an alphabet, which simply means it is a set, which
might be finite or infinite in this paper. An element a∈Γ is called a letter
and an element uin the free monoid Γ∗is called a word. A non-empty word
can be written as u=a1an, where ai∈Γ and n≥0. The number nis then
called the length of uand denoted u. The empty word has length 0 and is
denoted 1, as is customary for the neutral element in monoids or groups.
A crucial, but elementary fact for free monoids is that transposition is
equal to conjugacy. More precisely, in free monoids f z =zg implies that
f=rs,g=sr, and z=r(sr)mfor some m≥0. Essentially this implies a
straightforward algorithm for the conjugacy problem in free groups: on input
elements fand gof a free group first do cyclic reductions, to cyclically reduce
fand g. This costs only linear time. Then check whether the cyclically
reduced words fand gare transpose by searching for the word fas a factor
of the word g2. This is possible in linear time by a well-known pattern
matching algorithm, usually attributed to Knuth-Morris-Pratt [17], although
it was described earlier by Matiyasevich [22].
Frequently, sets and monoids come with an involution. An involution on
a set Xis a permutation aasuch that a=a. An involution of a monoid
4
satisfies in addition xy =y x. If the monoid is a group Gthen we always
assume that the involution is given by the inverse, thus g=g−1for group
elements. If the alphabet Γ has an involution, then it is extended to Γ∗by
defining a1an=ana1for ai∈Γ and n≥0. From now on we always assume
that Γ is equipped with an involution ∶Γ→Γ. Since the identity idΓis an
involution, this is no restriction.
2.2 Rewriting systems
Monoids and groups can be defined through a set of monoid generators Γ
and a set of defining relations S⊆Γ∗×Γ∗. A subset S⊆Γ∗×Γ∗is called a
semi-Thue system, or a string rewriting system. Given S, we define a relation
Ô⇒
S, called a one-step rewriting relation, on Γ∗by uÔ⇒
Svif and only if
u=pℓq and v=prq for some (ℓ, r ) ∈ S.
Let Xbe any set and Ô⇒⊆X×Xbe a relation. The iteration of at most
ksteps of Ô⇒ is denoted by ≤k
Ô⇒ while the reflexive and transitive closure
of Ô⇒ is denoted by ∗
Ô⇒. We also write x⇐Ô yand x∗
⇐Ô yto denote
yÔ⇒ xand y∗
Ô⇒ x, respectively. The reflexive, symmetric and transitive
closure of Ô⇒ is denoted by ∗
⇐⇒. Elements x∈Xsuch that there is no y
with xÔ⇒ yare called irreducible. The relation Ô⇒ is called:
1. strongly confluent, if y⇐Ô xÔ⇒ zimplies y≤1
Ô⇒ w≤1
⇐Ô zfor some w;
2. confluent, if y∗
⇐Ô x∗
Ô⇒ zimplies y∗
Ô⇒ w∗
⇐Ô zfor some wand
3. Church-Rosser, if y∗
⇐⇒ zimplies y∗
Ô⇒ w∗
⇐Ô zfor some w.
The following facts are well-known and easy to prove, see e.g. [2, 15].
1. Strong confluence implies confluence.
2. Confluence is equivalent to Church-Rosser.
A relation Ô⇒⊆X×Xis called terminating (or Noetherian), if there is
no infinite chain
x0Ô⇒ x1Ô⇒ ⋯ xi−1Ô⇒ xiÔ⇒ ⋯
5
For a semi-Thue system Sthe equivalence relation ∗
⇐⇒
Sis a congruence,
hence the equivalence classes form a monoid which is denoted by Γ∗S. This
is the quotient of the monoid Γ∗when Sis viewed as a set of defining relations.
We also say that Sis confluent, terminating etc., whenever Ô⇒
Shas the
corresponding property.
The main interest in a terminating and confluent system Sstems from
the fact that these properties (together with some other natural condition
on the computability of the one-step rewriting process) yield a procedure to
solve the word problem in the quotient monoid Γ∗S. If Γ is finite, then
decidability of the word problem is equivalent to the ability to compute
shortlex normal forms: first we endow the alphabet Γ with a linear order ≤.
The shortlex normal form for an element gin a quotient monoid Γ∗Sis then
the lexicographically first word among all geodesic words u∈Γ∗representing
g∈M. Recall, that a word u∈Γ∗is called a geodesic, if uhas minimal length
among all words representing the same element as uin Γ∗S.
Example 2.1. If the involution on Γis without fixed points, then we
can write Γas a disjoint union Γ=Σ˙
∪Σ. Then the rewriting system S=
{aa Ð→ 1a∈Γ}is strongly confluent and terminating; and the quotient
monoid Γ∗Sdefines the free group F(Σ). In this case geodesics are unique.
2.3 Thue systems
A semi-Thue system Sis called a Thue system, if Sdoes not contain any
length increasing rules and all length preserving rules are symmetric. This
means (ℓ, r) ∈ Simplies ℓ ≥ rand that ℓ = rimplies (r, ℓ) ∈ S, too.
The set Sof a Thue system splits naturally into two parts S=R˙
∪T, where
Rcontains the length reducing rules and Tcontains the symmetric length
preserving rules. In particular, R∩R−1=∅and T=T−1, where, as usual,
P−1= { (y, x) (x, y) ∈ P}for any relation P.
A Thue system Sis called geodesic, if starting from any word uand
applying only length decreasing rules we eventually obtain a geodesic word
v(a shortest word in the set {vu∗
⇐⇒
Sv}). Thus, we have u∗
Ô⇒
Rvfor some
geodesic word v.
A confluent, geodesic, Thue system is called geodesically perfect. This
means whenever u∗
⇐⇒
Sv, then we can first compute geodesics u∗
Ô⇒
R̂uand
v∗
Ô⇒
R̂v, by applying length reducing rules, and then we can transform ̂uinto
̂vby symmetric rules from T, that is ̂u∗
⇐⇒
T̂v(which in turn is equivalent
6
to ̂u∗
Ô⇒
T̂v). Thus the following statements are equivalent for geodesically
perfect systems.
1. u∗
⇐⇒
Sv.
2. ∃̂u, ̂v∶u∗
Ô⇒
R̂u∗
Ô⇒
T̂v∗
⇐Ô
Rv.
2.4 Cyclic words and cyclic rewriting
There are two principal ways of introducing cyclic words over an alpha-
bet Γ. The first one is based on combinatorics of words: in this case one
defines a cyclic word as an equivalence class of the transposition relation
on Γ∗. Thus, if w∈Γ∗then the cyclic word represented by wis the set
w∼= { vu ∈Γ∗uv =w}. The second one, defines the cyclic word repre-
sented by wto be the directed, Γ-labelled, cycle graph Cw, such that the label
of the cycle, when read with orientation, starting at an appropriate vertex, is
w. More precisely, if w=a1. . . an, n >0,then Cwis a directed graph with ver-
tices v1,...,vnand directed edges e1= (v1→v2),...,en−1= (vn−1→vn), en=
(vn→v1)where each edge eiis labelled by aiIn the graph-theoretic version,
an ordinary word w∈Γ∗can be viewed as a directed Γ-labelled path-graph
Pw: with vertices , v1,...,vn+1and edges e1= (v1→v2),...,en= (en→en+1)
with labels a1,...,an, respectively. If wis the empty word 1 then Pwand Cw
consist of a single vertex. We regard the combinatorial and graph theoretic
views of words and cyclic words as different aspects of the same objects and
pass from one to the other without further comment.
Graph rewriting (or transformation) is a well-established technique of
computing with graphs. We refer to the book [25] for details. In general, a
graph rewriting system consists of a set of graph rewriting rules of the form
(L, R), where Land Rare graphs. To apply such a rule to a given graph G
one finds a subgraph of Gisomorphic to Land replaces it by Raccording to
some prescribed procedure.
In our case the graphs Gare cycles Cw, where w∈Γ∗and the rewriting
rules are of the following two types:
1) (Pℓ, Pr)for some ℓ, r ∈Γ∗;
2) (Cℓ, Cr)for some ℓ, r ∈Γ∗,ℓ≠1.
Application of a rule (Pℓ, Pr)to a graph Cwinvolves replacing some path
subgraph Pℓof Cwby the path Pr. This can be clearly visualised as in
Figure 1. Application of the rule (Cℓ, Cr)to Cwis straightforward: if Cw
7
ℓ
○
Ô⇒
(ℓ,r)r
Figure 1: Cyclic rewriting when (ℓ, r) ∈ Sand ℓappears on the cycle.
is isomorphic to Cℓ(as a directed, labelled graph) then replace Cwby Cℓ.
Otherwise the rule does not apply. Clearly, the result of applying one of these
rules to cyclic word is a cyclic word. A rewriting system on cyclic words is
a set Tof rules of the type 1) and 2). We write CuÔ⇒
TCvif Cvcan be
obtained from Cuby one of the rules from T. In this case we may also write
u∼
○
Ô⇒
Tv∼or u○
Ô⇒
Tv. The definitions of Section 2.2 apply to an arbitrary
binary relation on a set X, and in particular to the relation Ô⇒
Ton the set of
cyclic words over Γ∗. Hence, we can talk about confluent, strongly confluent,
terminating, etc. rewriting systems on cyclic words.
The subsystem of Tconsisting of the rules of type 1) corresponds to a
string rewriting semi-Thue system S= {(ℓ, r) (Pℓ, Pr) ∈ T}. On the other
hand, let S⊆Γ∗×Γ∗be a semi-Thue system. Then Scomposed on the right
and left with the relation ∼defines a one-step relation ○
Ô⇒
Son cyclic words.
That is, we have u∼
○
Ô⇒
Sv∼, if and only if there are words u′and v′such that
u∼u′,u′Ô⇒
Sv′, and v′∼v. Obviously, if a rule (ℓ, r) ∈ Sis applied to u∼,
then the rewriting step u∼
○
Ô⇒
(ℓ,r)v∼may be understood as applying the rule
(Pℓ, Pr)to the graph Pu, as in Figure 1.
By analogy with string rewriting, we denote by ⊛
Ô⇒
Sthe reflexive and
transitive closure of ○
Ô⇒
S; write u○
⇐Ô
Svand u⊛
⇐Ô
Svfor v○
Ô⇒
Suand
v⊛
⇐Ô
Su, respectively; and denote by ⊛
⇐⇒
Sthe reflexive, symmetric, transitive
closure of ○
Ô⇒
S.
Neither confluence nor termination transfers from S(defined on words)
to ○
Ô⇒
S(defined on cyclic words).
Example 2.2. 1. Let Γ= {a, b, c, d}and let Sconsist of the following four
rules
abc Ð→ bac, cda Ð→ dca, bad Ð→ abd, dcb Ð→ cbd.
8
To see that Ô⇒
Sis confluent it is necessary to check all four cases where
the left-hand sides of rules overlap. For example the left-hand side of
abc Ð→ bac overlaps with the left-hand side of cda Ð→ dca. Therefore
we can rewrite abcda in two ways:
bacda ⇐Ô
Sabcda Ô⇒
Sabdca.
However
bacda Ô⇒
Sbadca Ô⇒
Sabdca,
so either way results in the same reduced word. The other three cases
are similar and so Ô⇒
Sis confluent. However, the cyclic rewriting
system defined by Sis not confluent. In fact abcd∼
○
Ô⇒
Sbacd∼and
abcd∼=bcda∼
○
Ô⇒
Sbdca∼. Both bacd∼and bcda∼are irreducible and
they are not equal.
2. Let Γ= {a, b}and S= {ba Ð→ ab2}. It is not difficult to see that
Ô⇒
Sis terminating. However the relation ○
Ô⇒
Son cyclic words is non-
terminating as
ba∼
○
Ô⇒
Sb2a∼
○
Ô⇒
Sb3a∼
○
Ô⇒
Sb4a∼
○
Ô⇒
S⋯.
A semi-Thue system Sis called C-confluent, if ○
Ô⇒
Sis confluent on cyclic
words. If Wis subset of cyclic words, then we also say that Sis C-confluent
on W, if ○
Ô⇒
Sis confluent on all cyclic words in W.
In the rest of the section we consider some general methods of transform-
ing confluent semi-Thue systems into C-confluent systems.
2.5 From confluence to cyclic confluence
Let S⊆Γ∗×Γ∗be a confluent semi-Thue system such that G=Γ∗Sis
agroup. In this section we consider the general question (in the spirit of a
Knuth-Bendix or Shirshov-Gr¨obner completion) of how to enlarge the system
Sby adding new rules in order to obtain another system ̂
Ssuch that the
following hold:
1. S⊆̂
Sand Γ∗S=Γ∗̂
S(i.e., ̂
Sis a conservative extension of S);
2. ○
Ô⇒
̂
S
is confluent on cyclic words (i.e., ̂
Sis C-confluent).
9
Usually, we refer to ̂
Ssatisfying 1 as an extension of S(omitting conserva-
tive). ̂
Ssatisfying 1 and 2 is termed a C-extension of S. Condition 1 ensures
that ̂
Sis still confluent (since S, and hence ̂
S, is Church-Rosser).
Now we fix a confluent semi-Thue system S⊆Γ∗×Γ∗such that G=Γ∗S
is a group. For each letter a∈Γ we can choose some fixed word ̃a∈Γ∗such
that ãa=1 in G. We extend this definition (in a unique way) to all words of
Γ∗as follows. Define ̃
1=1 and assume that ̃uhas been defined for all words
uof length at most n. Let u=va be a word of length n+1, with a∈Γ, v∈Γ∗.
Then define ̃u= ̃ãv. Clearly, ̃
̃w=win Gfor all words w∈Γ∗.
For x, y ∈Γ∗write x>y, if x∗
Ô⇒
Spyq with pq ≠1. Then >is a partial
order on Γ∗. Since x∗
Ô⇒
Sx=x1 (here 1 is the empty word) then x>1
for every non-empty word x. We call the system Sweakly-terminating if
the partial order >is well-founded, i.e., there are no infinite chains x1>
x2>x3>⋯. Clearly, if the system Sis terminating, then it is weakly-
terminating. Moreover, every semi-Thue system without length increasing
rules is weakly-terminating. Note that the empty word 1 is irreducible in
every weakly-terminating system. In particular, such a system does not have
rules of the type 1 →x̃xor 1 →̃xx, but x̃x∗
Ô⇒
S1 and ̃xx ∗
Ô⇒
S1 for any
x∈Γ∗, since Sis Church-Rosser and 1 is irreducible.
For a system Sdefine a semi-Thue system ̂
Sby the following rules uÔ⇒
̂
S
u′where:
1. uÔ⇒
Su′(original rule).
2. u=qv and u′= ̃prv, if exists pq Ð→ r∈S,p≠1≠q(prefix rule).
3. u=vp and u′=vr̃q, if exists pq Ð→ r∈S,p≠1≠q(suffix rule).
4. u′= ̃pr̃q, if exists puq Ð→ r∈S,p≠1≠q(infix rule).
It is clear that ̂
Ssatisfies S⊆̂
Sand Γ∗S=Γ∗̂
S. As before ⊛
Ô⇒
̂
S
denotes
the reflexive and transitive closure of ○
Ô⇒
̂
S
.
Theorem 2.3. Let S⊆Γ∗×Γ∗be a confluent weakly-terminating semi-Thue
system such that G=Γ∗Sis a group. Then the following hold:
1) uand vare conjugate in Gif and only if u⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
vfor some
(cyclic) word t.
2) the rewrite system ○
Ô⇒
̂
S
is confluent on (cyclic) words.
10
Proof. To prove 1) observe first (by inspection of all the rules in ̂
S) that
u⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
v(1)
(in fact, even u⊛
⇐⇒
̂
S
v) for some word t∈Γ∗implies that uand vare conjugate
in G.
Assume now that u, v ∈Γ∗define conjugate elements in G, i.e., xũx∗
⇐⇒
Sv
for some x∈Γ∗. We claim that in this case there exists t∈Γ∗for which (1)
holds. We proceed by Noetherian induction on x, i.e., by induction on the
number of predecessors of xrelative to >.
Since Sis Church-Rosser there exists w∈Γ∗such that xũx∗
Ô⇒
Sw∗
⇐Ô
Sv.
If xhas no predecessors then x=1 and the claim is obvious (in this case
t=w). Thus, we may assume that the claim holds for all y<x.
In the reduction xũx∗
Ô⇒
Swthe following cases may occur.
Case 1 (no overlap). Suppose one can factorise w=x′u′x′′ in such a way
that x∗
Ô⇒
Sx′,u∗
Ô⇒
Su′, and ̃x∗
Ô⇒
Sx′′, then we are done since x′′x′=1 in G,
so x′′x′∗
Ô⇒
S1 (1 is S-irreducible), and hence:
u∗
Ô⇒
Su′∗
⇐Ô
Su′x′′x′∼x′u′x′′ =w,
which proves the claim. Thus we may assume that there is no such factori-
sation.
Case 2 (overlap). Assume now that x∗
Ô⇒
Syp,u∗
Ô⇒
Sqv such that p≠
1, q ≠1 and pq Ð→ ris a rule of S. Then one has xũx=y(rṽp)̃y=vin
Gand y<x. Hence, by induction, rṽp⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
vfor some word t∈Γ∗.
Notice, that we can apply a prefix rule to qv and after a transposition obtain
u⊛
Ô⇒
̂
S
rṽp. Therefore, u⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
vand the claim holds.
The argument for the other possible overlap, when ̃x∗
Ô⇒
Sqy and u∗
Ô⇒
Svp,
is similar and we omit it.
Case 3 (nesting). We are left to consider the following situation: x∗
Ô⇒
S
yp,u∗
Ô⇒
Ss, and ̃x∗
Ô⇒
Sqz where p≠1≠qand psq Ð→ ris a rule of S. Again
xũx=y(r̃q̃p)̃yin Gand y<x. Hence, by induction, r̃q̃p⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
v,
for some word t. Applying an infix rule to sand a transposition yields
u⊛
Ô⇒
̂
S
r̃q̃p. The claim follows.
11
This finishes the proof of 1). Statement 2) follows from 1) since, as
mentioned above, u⊛
⇐⇒
̂
S
vimplies that uand vare conjugate in G.
Now we show that an extension S○(defined below) of Swhich is, in this
context, extremely natural is also a C-extension of S. Define S○to be the
extension of Sobtained by adding the rules 1 →ãaand 1 →̃aa, for every
a∈Γ. Thus
S○=S∪{1→ãa, 1→̃aa a∈Γ}
and, since G=Γ∗Sis a group, S○is indeed a C-conservative extension of S.
Theorem 2.4. Let S⊆Γ∗×Γ∗be a confluent weakly-terminating semi-Thue
system such that G=Γ∗Sis a group. Then the following hold:
1) uand vare conjugate in Gif and only if u⊛
Ô⇒
S○t⊛
⇐Ô
S○vfor some
(cyclic) word t.
2) S○is a C-extension of S.
Proof. If u⊛
Ô⇒
S○t⊛
⇐Ô
S○vfor some word tthen uand vare obviously conjugate
in G. Conversely, if uand vare conjugate in G, then by Theorem 2.3
u⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
vfor some word t. Observe, that application of a prefix, suffix,
or infix rule from ̂
Sis equivalent to a sequence of rule applications from S○,
so u⊛
Ô⇒
̂
S
t⊛
⇐Ô
̂
S
vimplies u⊛
Ô⇒
S○t⊛
⇐Ô
S○v. Now the result follows.
2.6 From strong to cyclic confluence in groups
The transformation of a semi-Thue system Sinto the larger system S○de-
scribed in Section 2.5 leads to length increasing rules. This is in some sense
unavoidable. Indeed, assume we have ab =cand ba =din the quotient
M=Γ∗Swhere c, d ∈Γ are letters. In general we cannot expect that
c=d∈M. But cand dare transpose, so we need cyclic rewriting rules to
pass from cto dor vice versa. If we wish to do this by string rewriting and
transpositions, then we are forced to pass from cto dvia cyclic words of
length at least 2. This is what happens in building ̂
Sand S○.
Another idea is to introduce special rules which directly rewrite short
cyclic words into each other, if they represent distinct conjugate elements.
In this case we have rules that rewrite cyclic words, but these rules are not
induced by any string rewriting rules in the system (via equivalence relation
∼). We now make this precise.
12
We start with a semi-Thue system S⊆Γ∗×Γ∗, which we allow to be
infinite. Define
m(S) = sup { ℓ (ℓ, r) ∈ S}.
We say that Sis left-bounded if m(S) < ∞. From now on we assume that
the empty word is S-irreducible and, to exclude trivial cases, that 2 ≤m(S).
To this end, we say that Sis a standard semi-Thue system, if it satisfies the
two conditions above: that is
1. (1, r) ∉ S, for all non-empty words r∈Γ∗, and
2. 2 ≤m(S) < ∞.
A cyclic word w∼is called S-short if w ≤ 2m(S)−2, and it is called
strictly S-short if w < 2m(S)−2. When Sis fixed we refer to such words
simply as short or strictly short.
In the following let C(S)denote any relation defined on the set of cyclic
words which satisfies the following two conditions:
C1 If u∼
○
Ô⇒
Sv∼, then (u∼, v∼) ∈ C(S), i.e., ○
Ô⇒
S⊆C(S).
C2 If (u∼, v∼) ∈ C(S), then uand vare conjugate in Γ∗S.
Later, we discuss the possibility of constructing relations C(S)with these
properties. We write u○
Ô⇒
C(S)vand v○
⇐Ô
C(S)uif (u∼, v∼) ∈ C(S)or if u∼=
v∼. (Thus, both ○
Ô⇒
C(S)and ○
⇐Ô
C(S)are reflexive.) Moreover, we use ⊛
Ô⇒
C(S)and
⊛
⇐⇒
C(S)again, for the transitive, and for the symmetric and transitive closure,
respectively, of ○
Ô⇒
C(S). As C(S)is a relation on cyclic words, when we say C(S)
is confluent, or strongly confluent, unless we explicitly specify an alternative,
we mean confluent or strongly confluent on the set of all cyclic words.
Theorem 2.5. Let S⊆Γ∗×Γ∗be a standard strongly confluent semi-Thue
system such that C(S)satisfies the two conditions C1 and C2 above. Then
the following assertions are equivalent:
1.) The system C(S)is confluent.
2.) The system C(S)is confluent on all short cyclic words w. (That is if w
is short and u⊛
⇐Ô
C(S)w⊛
Ô⇒
C(S)vthen there exists tsuch that u⊛
Ô⇒
C(S)t⊛
⇐Ô
C(S)v.)
13
Proof. We have to show only that if C(S)is confluent on all short cyclic
words, then C(S)is confluent.
First consider u○
⇐Ô
C(S)w○
Ô⇒
C(S)vwhere w ≥ 2m(S)−1. Then the two rules
applied to the cyclic word w∼are inherited from the semi-Thue system S.
Since wis long enough the corresponding left-hand sides overlap in the cyclic
word wat most once. Since Sis strongly confluent, we see that there is some
cyclic word tsuch that
u○
Ô⇒
C(S)t○
⇐Ô
C(S)v.
Next, consider
u=wk
○
⇐Ô
C(S)⋯○
⇐Ô
C(S)w0
○
Ô⇒
C(S)v1
○
Ô⇒
C(S)⋯○
Ô⇒
C(S)vm=v.
We may assume that m≥k≥1. We perform an induction on (k, m)in the
lexicographical order.
If none of w0, . . . , wk−1is short, then by strong confluence of Swe have
the following situation.
u○
Ô⇒
C(S)w′
k
○
⇐Ô
C(S)⋯○
⇐Ô
C(S)w′
1
○
⇐Ô
C(S)v1
⊛
Ô⇒
C(S)v.
Thus, we are done by induction on m. Therefore let wℓbe a short cyclic
word where ℓ≤k−1. By induction on kwe see that there exists
wℓ
⊛
Ô⇒
C(S)t⊛
⇐Ô
C(S)v.
Moreover, u⊛
⇐Ô
C(S)wℓand C(S)is confluent on wℓbecause wℓis a short cyclic
word. Hence we find
u⊛
Ô⇒
C(S)t′⊛
⇐Ô
C(S)t⊛
⇐Ô
C(S)v.
Corollary 2.6. Let S⊆Γ∗×Γ∗be a standard strongly confluent semi-Thue
system such that Γ∗Sis a group and such that first, C(S)is confluent on
all short cyclic words and second, it satisfies the two conditions C1 and C2
above. Then two words uand vare conjugate in Γ∗Sif and only if there
exists a cyclic word tsuch that
u∼
⊛
Ô⇒
C(S)t⊛
⇐Ô
C(S)v∼.
14
Proof. Clearly, u∼
⊛
⇐⇒
C(S)v∼implies conjugacy. Now, if uand vare conjugate,
then there is some xsuch that xux−1∗
⇐⇒
Sv. This implies xux−1
∼
⊛
⇐⇒
C(S)v∼. We
have xx−1∗
Ô⇒
S1, because Sis standard and confluent, hence xux−1
∼
⊛
Ô⇒
C(S)u∼.
We conclude u∼
⊛
⇐⇒
C(S)v∼. The result follows by Theorem 2.5.
2.7 A Knuth-Bendix-like procedure on cyclic words
If a system C(S), satisfying C1 and C2 above, is large enough to ensure
u∼
⊛
Ô⇒
C(S)v∼whenever u∼and v∼are conjugate in Γ∗Swith ushort, then
we can apply Theorem 2.5; and we can use the system C(S)for solving
conjugacy in Γ∗S. In order to construct such a system we may use a form
of Knuth-Bendix completion. This can be done in a very general way; which
is fairly standard but technical, if we work out all details. Here we wish to
restrict an analogue of Knuth-Bendix completion to short words; for which
we need some additional hypotheses.
We assume throughout this section that the alphabet Γ is well-ordered
by <. We extend this well-order to the shortlex order <on Γ∗as usual:
we write u<vif either u < vor u = vand u=pax,u=pby with
a, b ∈Γ such that a<b. Moreover, we extend the well-order to cyclic words
by representing a cyclic word w∼by the minimal shortlex word in its class
w∼= { uv vu =w}. Hence, there is well-order on the set of cyclic words.
For any relation R⊆Γ∗×Γ∗we define the descending part of Rto be
̃
R= {(l, r) ∈ R∶l>rin the shortlex ordering}.
The new restriction we put on Sis that we assume that, for all (ℓ, r) ∈ S,
we have r ≤ ℓ. In particular, if wis short and w⊛
Ô⇒
Sv, then vis short, too.
Now let C(S)satisfy C1 and C2 above. We say that (u∼, v∼) ∈ C(S)is a
short critical pair, if u∼>v∼(in the shortlex ordering) and for some S-short
word wwe have:
u∼
○
⇐Ô
C(S)w∼
○
Ô⇒
C(S)v∼(2)
We say that the critical pair (u∼, v∼)in (2) is shortlex resolved, if
u∼
⊛
Ô⇒
̃
C(S)
t∼
⊛
⇐Ô
̃
C(S)
v∼,
for some twith v≥t(where ̃
C(S)is the descending part of C(S)). By
resolving (u∼, v∼)we mean adding the rule (u∼, v∼)to C(S). (Note that, by
definition, u∼>v∼.) Hence by resolving we force (u∼, v∼)to be resolved.
15
If we begin by taking C(S)equal to ○
Ô⇒
Sthen, by resolving short pairs,
we may form new systems which still satisfy C1 and C2. If the alphabet Γ
is finite, then this procedure of adding more rules terminates because there
are only finitely many short words. In general, there exists a limit system
C∗(S), satisfying C1 and C2 and such that all short critical pairs are shortlex
resolved, but if Γ is infinite then we may only have a semi-procedure for its
construction.
Theorem 2.7. Let S⊆Γ∗×Γ∗be a standard strongly confluent semi-Thue
system such that Γ∗Sis a group and such that, for all (ℓ, r) ∈ S, we have
r ≤ ℓ. Let C∗(S)be constructed as above by resolving short critical pairs.
Then the following two assertions hold:
1. The system C∗(S)is standard and confluent.
2. Two words uand vare conjugate in Γ∗Sif and only if there exists a
cyclic word t∼such that
u∼
⊛
Ô⇒
C∗(S)t∼
⊛
⇐Ô
C∗(S)v∼.
Proof. By construction C∗(S)is standard. Having shortlex resolved all short
critical pairs, the descending part ̃
C=
C∗(S)of C∗(S)is terminating and
contains all new rules (u∼, v∼)added to the system ○
Ô⇒
S. Therefore ̃
Cis
locally confluent on short words. Moreover, if u∼
○
Ô⇒
Sv∼then, since <is a
total order and l ≥ rfor all (l, r ) ∈ S, either (u∼, v∼)or (v∼, u∼)belongs to
̃
C. Hence u∼
⊛
⇐⇒
C∗(S)v∼if and only if u∼
⊛
⇐⇒
̃
C
v∼. (Note that we don’t claim that
̃
Csatisfies C1 or C2. We don’t even have S⊆̃
C, in general.)
We are now ready to show that C∗(S)is confluent on short words. Con-
sider the following situation where wis short:
u∼
⊛
⇐Ô
C∗(S)w∼
⊛
Ô⇒
C∗(S)v∼.
Since r ≤ ℓfor (ℓ, r) ∈ S(and hence for all (ℓ, r) ∈ C∗(S)) we see that u
and vare short, and moreover u∼
⊛
⇐⇒
̃
C
v∼. Note that the path u∼⇐⇒
̃
C
u′
∼⇐⇒
̃
C
⋯ ⇐⇒
̃
C
v∼( via w∼) never leaves the set of short words. Being terminating
and locally confluent, the system ̃
Cis confluent on short words. Hence, since
u∼
⊛
⇐⇒
̃
C
v∼, there exists a cyclic word t∼such that
u∼
⊛
Ô⇒
̃
C
t∼
⊛
⇐Ô
̃
C
v∼.
16
As ̃
C⊆C∗(S), we see that C∗(S)is confluent on short words, as claimed.
Finally, C∗(S)satisfies conditions C1 and C2 above; and so Corollary 2.6
applies, to give the result.
2.8 Strongly confluent Thue systems
If the system Sis Thue (c.f. Section 2.3) then we may construct C∗(S)
in finitely many steps as follows. We start with C0=C0(S) = ○
Ô⇒
S. This
is a relation defined on the set of cyclic words where all rules are either
length decreasing or length preserving and then symmetric. We call any
such relation on cyclic words Thue.
At each step let us define a Thue relation Cisatisfying conditions C1 and
C2 above. We let Uibe the set of “unresolved short critical pairs” (u∼, v∼),
which are defined in the Thue case as follows:
u∼
○
⇐Ô
Ci
w∼
⊛
Ô⇒
Ci
w′
∼
○
Ô⇒
Ci
v∼
where wis S-short, w = w′≥ u ≥ v ≥ 1, and neither u∼
⊛
Ô⇒
Ci
v∼nor
u∼
⊛
⇐Ô
Ci
v∼.
Note that, since w = w′we have u∼
○
⇐Ô
Ci
w∼
⊛
⇐Ô
Ci
w′
∼
○
Ô⇒
Ci
v∼, too. Thus,
for unresolved pairs we must have w > u ≥ v ≥ 1. (Because if, say w = v,
then u∼
⊛
⇐Ô
Ci
w′
∼
○
⇐Ô
Ci
v∼.)
At the next step we let Ci+1be the relation obtained from Ciby adding
a pair (u∼, v∼)to Ci, for all (u∼, v∼) ∈ Ui, and, in addition, by adding (v∼, u∼)
whenever u∼ = v∼. This keeps Ci+1Thue. Finally, we let
C∗(S) = ⋃{Ci(S) i∈N}.(3)
Theorem 2.8. Let Sbe a standard, strongly confluent, Thue system, let
m=m(S)and let C∗(S)be the system defined in (3) above. Then C∗(S) =
C2m−2, and C∗(S)is a confluent, Thue system, satisfying conditions C1 and
C2.
Proof. By definition Ciare Thue for all i≥0; and short words have length
at most 2m−2. When considering u∼
○
⇐Ô
Ci
w∼
⊛
Ô⇒
Ci
w′
∼
○
Ô⇒
Ci
v∼we may assume
that u < w(see above) and that u∼
○
⇐Ô
Ci∖Ci−1
w∼(or w′
∼
○
Ô⇒
Ci∖Ci−1
v∼). Thus,
at every step the words wunder consideration get shorter. We conclude
C∗(S) = C2m−2(S), as claimed.
17
Next, we show that C∗(S)is confluent on short cyclic words. To this end
we define an equivalence relation ≡on cyclic words by u≡vif u∼
⊛
Ô⇒
C∗(S)v∼and
u∼
⊛
⇐Ô
C∗(S)v∼. Thus, if u≡vthen u∼
⊛
⇐⇒
C∗(S)v∼and u = v. We can view C∗(S)
as a terminating rewriting system on equivalence classes [u]={vv≡u}.
By construction, C∗(S)is locally confluent on classes [w], where wis short.
But together with termination, we see that C∗(S)is actually confluent on
these classes [w]. But this implies that C∗(S)is confluent on short cyclic
words, because it is Thue. Finally, C∗(S)satisfies the two conditions C1 and
C2 above. Since Sis also a standard, strongly confluent semi-Thue-system,
we may apply Theorem 2.5.
2.9 Cyclic geodesically perfect systems
In this section we consider an analogue for cyclic rewriting systems of geodesi-
cally perfect string rewriting systems; and adapt our Knuth-Bendix comple-
tion process to these systems. Let S⊆Γ∗×Γ∗be a standard semi-Thue
system such that Γ∗Sis a group. A cyclic word w∼is called geodesic (w.r.t.
S), if wis a shortest word in its conjugacy class. That is
w=min uu∈Γ∗and ∃x∶xux−1=w∈Γ∗S.
A cyclic word w∼is called quasi-geodesic (w.r.t. S), if it is either geodesic
or it is strictly S-short, but it is not equal to the neutral element in Γ∗S.
Note that all non-trivial geodesic cyclic words are quasi-geodesic and more
importantly in 2-monadic systems every quasi-geodesic cyclic word is actually
geodesic.
Now, a Thue relation C(S)on cyclic words, satisfying C1 and C2 above,
is called quasi-geodesic, if by applying a sequence of length reducing rules
from C(S)to a cyclic word w∼we eventually derive a quasi-geodesic cyclic
word u∼. In order to be geodesically perfect C(S)must satisfy stronger
conditions: C(S)is called geodesically perfect if, by applying a sequence of
length reducing rules from C(S)to a cyclic word w∼, we eventually derive a
geodesic cyclic word u∼. Moreover, if two geodesics u∼and v∼can both be
derived from w∼, then it must be possible to rewrite u∼into v∼using only
length preserving rules from C(S). Note that every geodesically perfect Thue
system on cyclic words is confluent.
Now, if S⊆Γ∗×Γ∗is a Thue system then we say that Sis C-quasi-geodesic
if the system ○
Ô⇒
S, on cyclic words, is quasi-geodesic. The following result
shows that a geodesic Thue system is innately C-quasi-geodesic.
18
Theorem 2.9. Let S⊆Γ∗×Γ∗be a standard, geodesic, Thue system. Then
Sis C-quasi-geodesic.
Proof. We have to show the following: if u∼
⊛
⇐⇒
Sw∼and u<w, then either
a length reducing rule applies to the cyclic word w∼or w∼is strictly S-short.
To begin with let u∼
⊛
⇐⇒
Sw∼. Then there is a sequence u=w0,...,wℓ=w
such that wi−1and wiare related in one of the following three ways:
wi−1Ô⇒
Swior wi−1⇐Ô
Swior wi−1∼wi.
First, we claim that there exist m∈Nand u1, u2∈Γ∗such that u1uk−mu2
∗
⇐⇒
S
wk, for all k>m.
This is true for ℓ=0 with m=0. For ℓ≥1 the result holds by induction
for v=w1,...,wℓ=wwith some m∈Nand v1, v2∈Γ∗. Now, if uÔ⇒
Sv, then
we have v1uk−mv2
∗
Ô⇒
Sv1vk−mv2
∗
⇐⇒
Swk, for all k>m. Similarly, if u⇐Ô
Sv,
then we have v1uk−mv2
∗
⇐Ô
Sv1vk−mv2
∗
⇐⇒
Swk, for all k>m. Now, let u=u2u1
and v=u1u2. Define m′=m+1. We have:
v1u1uk−m−1u2v2
∗
⇐⇒
Sv1vk−mv2
∗
⇐⇒
Swk,for all k>m.
Replacing m,u1and u2with m′,v1u1, and u2v2, respectively, we see that
the claim holds.
Next, assume that we have u<wand choose m,u1and u2as above.
Take klarge enough to make wk>u1uk−mu2. Since u1uk−mu2
∗
⇐⇒
Swkand
Sis geodesic, a length reducing rule (ℓ, r )∈Sapplies to wk. If ℓ≤w, then
the same rule applies to the cyclic word w∼, and we are done. In the other
case, wis strictly S-short, and we are done, too.
In the next section of the paper we shall be concerned with standard,
geodesically perfect, Thue string rewriting systems S, which are 2-monadic:
that is m(S)=2. For the rewriting system ○
Ô⇒
Sinduced by such S, there
is a particularly simple form of Knuth-Bendix completion. In this case we
consider an short critical pair (u∼, v∼)to be “unresolved” if it arises from the
situation
u∼
○
⇐Ô
Sw∼
○
Ô⇒
Sv∼,(4)
where wis short and w>u≥v≥1. We resolve the short critical pair
of (4) by adding the rules (u∼, v∼)and (v∼, u∼). Let C†(S)be the system
obtained from ○
Ô⇒
Sby resolving all short critical pairs of the form (4). Note
19
that if (u∼, v∼)is a short critical pair then both uand vare strictly short
and non-trivial so, Sbeing 2-monadic, we have u=v=1.
Corollary 2.10. Let S⊆Γ∗×Γ∗be a standard, 2-monadic, geodesically
perfect, Thue system, such that Γ∗Sis a group, and C†(S)is confluent.
Then C†(S)satisfies C1 and C2 and is geodesically perfect. Moreover two
cyclic words u∼and v∼are conjugate in Γ∗Sif and only if there exists a
cyclic word t∼such that
u∼
⊛
Ô⇒
C†(S)t∼
⊛
⇐Ô
C†(S).
Proof. By construction C†=C†(S)satisfies C1 and C2. Two elements u, v ∈
Γ∗are conjugate if and only if u∼
⊛
⇐⇒
C†v∼; so the final statement holds if C†is
confluent. Therefore it is sufficient to prove that C†is geodesically perfect.
Consider w⊛
Ô⇒
C†vsuch that vhas minimal length with this property (so
is geodesic) and let w⊛
Ô⇒
C†ube some maximal derivation using only length
reducing rules from the cyclic rewriting system C†. Clearly, u≥v; and
Theorem 2.9 implies that Sis C-quasi-geodesic so either u=vor uis
strictly S-short. We have to show that we can transform uinto vby length
preserving rules from C†. This is clear, if vis not strictly S-short, because
then u=v, and C†is confluent and Thue. For m(S)=2, a strictly S-short
word vis either a letter or the empty word 1. But if v=1 we have w∗
Ô⇒
Sv
because Sis a confluent semi-Thue system and 1 is irreducible ; and it follows
from the definitions of ⊛
Ô⇒
C†and uthat u=1 as well. There remains the case
v∈Γ. Since Sis C-quasi-geodesic we have u=1, too. As C†is confluent
and Thue we can transform the letter uinto v, by applying length preserving
rules of C†.
3 Stallings’ pregroups and their universal groups
We now turn to the notion of pregroup in the sense of Stallings, [27], [28]. A
pregroup is a set Pwith a distinguished element ε, equipped with a partial
multiplication (a, b)↦ab which is defined for (a, b)∈D, where D⊆P×P,
and an involution a↦a, satisfying the following axioms, for all a, b, c, d ∈P.
(By “ab is defined” we mean that (a, b)∈D.)
(P1) aε and εa are defined and aε =εa =a;
(P2) aa and aa are defined and aa =aa =ε;
20
(P3) if ab is defined, then so is ba, and ab =b a;
(P4) if ab and bc are defined, then (ab)cis defined if and only if a(bc)is
defined, in which case
(ab)c=a(bc);
(P5) if ab, bc, and cd are all defined then either abc or bcd is defined.
It is shown in [11] that (P3) follows from (P1), (P2), and (P4), hence can be
omitted.
For a, b ∈Pwe write ab ∈P, to mean that ab is defined. Also we use the
notation [ab]to indicate that ab ∈Pand, under the partial multiplication,
(a, b)↦[ab]. This notation is extended recursively to products of more than
two elements of P: if w∈P∗, where the notation has been established for
words shorter than w, and whas a factorisation w=uv, such that u, v ∈P
and [u][v]is defined, we write w∈Pand use [w]to denote the product
[u][v]∈P. (Note though that, for example, [abc]means only that one of
[ab]cor a[bc]belongs to P. (cf. Lemma 3.2.))
The set Pcan be considered as a possibly infinite alphabet. The axioms
above lead to the following definitions of Thue systems Sε,S(P)and the
universal group U(P).
Definition 3.1. 1. The system Sε⊆P∗×P∗is defined by the following
rules: εÐ→ 1(= the empty word)
ab Ð→ [ab]if (a, b)∈D
ab ←→[ac][cb]if (a, c),(c, b)∈D
2. Let Γ=P∖{ε}. The system S(P)⊆Γ∗×Γ∗is defined as follows:
ab Ð→ 1if (a, b)∈Dand [ab]=ε.
ab Ð→ [ab]if (a, b)∈Dand [ab]≠ε.
ab ←→[ac][b]if (a, c),(c, b)∈D, and (a, b)∉D.
We say that S(P)is the Thue system associated with P.
3. The universal group U(P)of a pregroup Pis the group
U(P)=Γ∗{ℓ=r (ℓ, r)∈S(P) } .
Tietze transformations may be applied to the presentation P∗Sεto give
the presentation Γ∗S(P); so U(P)≅P∗Sε.
Areduced word is an element p1⋯pnof P∗such that all pi∈Γ and [pipi+1]∉
P, for ifrom 1 to n−1.
21
The relationships between a pregroup, these rewriting systems and the
universal group rest on several key lemmas, the most important of which we
restate here for completeness.
Lemma 3.2 ([27]).Let a, b, c, d, g, h ∈P.
1.) If ab ∈Pthen [ab]b∈Pand [abb]=a.
2.) If ab ∉Pbut ac and cb ∈Pthen [ac][cb]∉P.
3.) If abc is a reduced word and ad, db ∈Pthen [ad][db]cis a reduced word.
4.) If ab ∉Pbut ac,cb,bd ∈Pthen cbd ∈P. (That is [cb]d∈Pfrom which
it follows that [c[bd]] =[[cb]d].)
5.) If g b, bh, gbc, cbh ∈P, but gh ∈Pthen bc ∈P.
Proof. 1.) Apply (P4) to the triple a, b, b.
2.) Use 1.) and apply (P4) to the triple [ac],cand b.
3.) From the above [ad][db]is reduced and ddb ∈P. If dbc ∈Pthen
consider the four element product ad[db]c. From (P5), either ab ∈Por
bc ∈P, a contradiction.
4.) Consider the four elements [ac],c,band d, of P. The product of each
adjacent pair is defined, so (P5) implies either ab =[ac][cb]∈P, or
cbd ∈P.
5.) Consider the product of four elements g[gb]c[cbh]. By hypothesis we
have gbc, bh ∈P. Moreover, [gb]c[cbh]=gh ∉P. Hence, by (P5) we
conclude g[gb]c=[bc]∈P.
As a consequence of Lemma 3.2.3.) and 4.) the set of reduced words
coincides with the set of S(P)-geodesic and the set of Sε-geodesic words.
The length preserving rule ←→of Sεis the length 2 case of Stallings’
interleaving relation ≈defined on words in P∗as follows. If ai, ci∈P, for
i=1,...n, and ci−1ai,aiciand ci−1aici∈Pwith c0=cn=ε, then
a1⋯an≈b1⋯bn,
where bi=[ci−1aici]. Stallings used Lemma 3.2 to show that interleaving
is an equivalence relation on reduced words and this equivalence relation is
central to the proof of Theorem 3.4.1) in [27]. Another approach is taken in
[7], based on the following lemma, which is again proved using Lemma 3.2.
22
Lemma 3.3. The Thue system Sεis strongly confluent.
Parts 1) and 2) of the following theorem are from Stallings [27]. Part 3)
is from [7].
Theorem 3.4 ([27],[7]).Let Pbe a pregroup. Then the following hold.
1) Pembeds into U(P).
2) If gand hare reduced words P∗then g=U(P)hif and only if his an
interleaving of g.
3) S(P)is a geodesically perfect Thue system.
Proof. 1) is a direct consequence of Lemma 3.3 and the remark following the
proof of Proposition 3.6. 2) follows from 3) and Lemma 3.2. The proof of 3)
is given in [7]: however, for completeness we give a proof. Consider a word
u=a1⋯anwith ai∈Γ such that aiai+1is not defined in Pfor 1 ≤i<n. Assume
that after a sequence of applications of symmetric rules, we can apply a length
reducing one. We have to show that some length reducing rule applies to u.
We may assume that the sequence of applications of symmetric rules is not
empty, but as short as possible. The corresponding word contains a factor
abcd with a, b, c, d ∈Γ and neither ab,bc nor cd defined in P. Applying
the last symmetric rule yields a[bx][xc]d. The length reducing rule cannot
then apply to [bx][xc], since this is not defined, by Lemma 3.2.2.), and so
must apply to a[bx]or [xc]d. In both cases we have a contradiction to
Lemma 3.2.3.).
Remark 3.5. Every group Gis the universal pregroup of some pregroup P.
Indeed, G=U(G). Moreover, Theorem 3.4 tells us that every pregroup P
can be defined as a subset P⊆Ginside a group Gsuch that 1∈P,a∈P
implies a−1∈P, and Psatisfies the axiom (P5). Having such a subset the
domain Dbecomes D={(a, b)∈P×Pab ∈P}.
3.1 Amalgamated products and HNN-extensions
The guiding example of an universal group in the sense of Stallings is the
amalgamated product G=A∗HBof two groups over a common subgroup
H=A∩B. In this case P=A∪Bforms a pregroup with U(P)=G. In
this case, for a, b ∈P, the product ab is defined in Pif and only if a, b ∈Aor
a, b ∈B. The verification of (P5) is straightforward.
The other obvious example of an universal group is the case where G=
HNN(H, t;t−1At =B)is an HNN-extension over two isomorphic subgroups
23
A, B in some base group H. (That is there is an isomorphism ϕ∶AÐ→ Band
“t−1At =B” denotes the set of relations of the form t−1at =aϕ, for all a∈A.)
In this case we can choose P=H∪H t−1H∪H tH. Again, the verification of
(P5) is straightforward.
3.2 Fundamental groups of graph of groups
The notion of the fundamental group of a graph of groups generalises amal-
gamated product and HNN-extension to a much broader class. The concept
of a graph of groups is due to Serre and the development of Bass-Serre theory
has been a major achievement in modern group theory. We refer to the books
[26], [1], and to [24] for the background.
Avirtually free group is a group Ghaving a free subgroup of finite index.
They are related to graphs of groups as follows.
Proposition 3.6. Let Gbe a finitely generated group. The following condi-
tions are equivalent.
1. Gis the fundamental group of a finite connected graph of groups where
all vertex groups are finite.
2. Gis the universal group of some finite pregroup.
3. Gcan be presented by some finite geodesic system.
4. Gis virtually free.
Propostion 3.6 is taken from [7, Cor. 8.7] and combines several results
from the literature. It follows from [24], [7], [16], and [23].
4 Conjugacy in universal groups
We shall apply Theorem 2.5 and Corollary 2.10 to the universal group of
a pregroup and in particular to the conjugacy problem. For this we fix
a pregroup P, we let U(P)be its universal group; and denote by Sεand
S=S(P)the Thue systems of Defintion 3.1. Let C†(Sε)and C†(S)be the
cyclic rewriting systems defined by resolving short critical pairs in the sense
of Section 2.9.
Acyclically reduced word is a cyclic word over Γ∗which is geodesic with
respect to the rewriting system C†(S). We also refer to words w∈w∼as
cyclically reduced if w∼is cyclically reduced. In particular all elements of Γ
are cyclically reduced.
24
Lemma 4.1. Let g∈P∗be a cyclically reduced word and let h∈P∗be a word
such that h∼is obtained from g∼by applying a sequence of length preserving
rules of C†(Sε). Then h∼is cyclically reduced and h=g.
Proof. By induction it is enough to prove the case where h∼is obtained from
g∼by applying a single rule. If g∈Pthen g≠ε, as εis not cyclically reduced,
so g∈Γ, and the result follows.
If g=n≥2 then there exists a word g1⋯gn∈g∼and an element c∈P
such that gigi+1∉Pfor all i(subscripts modulo n), g1c∈P,cg2∈Pand f=
[g1c][cg2]⋯gn∈h∼. As g1⋯gng1⋯gnis reduced, it follows from Lemma 3.2,
2.) & 3.) that f2is reduced. Therefore f, and so also h, is cyclically reduced,
as required.
This lemma suggests that cyclically reduced cyclic words under cyclic
rewriting should play the role of reduced words under standard rewriting.
This works as expected, with the exception of the behaviour of words of
length 1. From Theorem 3.4, two elements of Γ are equivalent under Sonly
if they are equal in Γ. However this is not true of cyclic words of length 1 and
the system C†(S), and we often have to treat words of length one separately
in what follows.
Let u=a1⋯an∈Γ∗with ai∈Γ. A cyclic permutation of uis any element
of u∼. Thus, a cyclic permutation is the same as a transposition in Γ∗. Let
n≥2. If for i=1,...n, there are elements bi, ci∈Psuch that ci−1ai,aiciare
in P, and bi=[ci−1aici](subscripts modulo n), then any element of v∼, where
v=b1⋯bnis called a cyclic interleaving of u; and u∼is also called a cyclic
interleaving of v∼. A preconjugation of uby c∈P(when n≥2) is the cyclic
interleaving v=[ca1]a2⋯an−1[anc].
For u∈Γ (i.e., n=1) a cyclic interleaving of uby c∈Pis defined as
v=[cuc]in case that cuc ∈Pis defined. A preconjugation is defined to be a
cyclic interleaving in this case.
In all cases every cyclic interleaving of umay be obtained by a cyclic per-
mutation, followed by an interleaving, followed by a preconjugation. More-
over every cyclic interleaving of uis conjugate to uin U(P). The following
lemma describes more precisely how these definitions are related.
Lemma 4.2. Let gand hbe cyclically reduced words over Γ∗. If g=1
then his a cyclic interleaving of gif and only if h∼is obtained from g∼by
applying a length preserving rule from C†(S). If g≥2, then the following
are equivalent.
1.) h∼is obtained from g∼by the application of a finite sequence of length
preserving rules from C†(S).
25
2.) There exists a word f, obtained from gby a cyclic permutation followed
by a single preconjugation, such that h=U(P)f.
3.) his a cyclic interleaving of g.
Proof. First consider the case n=1. Then his a cyclic interleaving of gif
only if there exists b∈Psuch that either bg or gb ∈Pand [bgb]=h∈P. On
the other hand, there is a symmetric rule in C†(S)transforming g∼to h∼if
and only if there exists b∈Psuch that either bg ∈Pand [bg]bÔ⇒
Sh(in
which case b[bg]Ô⇒
Sg); or gb ∈Pand b[gb]Ô⇒
Sh.
Now suppose n≥2. We show first that 3.) implies 1.). If 3.) holds then
there exist gi, ai∈Psuch that ai−1gi,giaiand ai−1giai∈Pand his a cyclic
permutation of h1⋯hn, where hi=[ai−1giai]. Therefore, we may successively
apply symmetric rules of Sto g∼to obtain (h1⋯hn)∼=h∼as required.
Next we show that 1.) implies 2.). If 1.) holds then there exist words
g0,...,gnin Γ∗such that g0=g,h∈gn∼and gi+1∼is obtained by applying
a symmetric rule of C†(S)to gi∼. If n=0 then his a cyclic permutation
of gand there is nothing further to do. Assume then that n>0. From
Lemma 4.1 giis cyclically reduced for all i. By definition there exists a word
g0=a1. . . an∈g∼and an element c∈Psuch that a1c∈P,ca2∈Pand
g1=b1⋯bn, where b1=[a1c],b2=[ca2]and bi=ai, for i≥2. By induction,
there exists a word f1, obtained from g1by a cyclic permutation followed
by a single preconjugation, such that h=U(P)f1. There are several cases to
consider, depending on which cyclic permutation of g1is taken. Assume f1
is a preconjugation of a cyclic permutation bi+1⋯biof g1, where 0 ≤i≤n−1.
That is, there exists d∈Psuch that dbi+1∈P,bid∈Pand f1=ci+1⋯ci, where
ci+1=[dbi+1],ci=[bid]and cj=bj, if j≠i, i +1. Thus
f1=
c1c2=[d[a1c]][[ca2]d]if i=0 and n=2
c1c2⋯cn=[d[a1c]][ca2]⋯[and]if i=0 and n≥3
c2c3⋯cnc1=[d[ca2]]a3⋯an[[a1c]d]if i=1
c3⋯cnc1c2=[da3]⋯an[a1c][[ca2]d]if i=2
ci+1⋯cnc1c2⋯ci=[dai+1]⋯an[a1c][ca2]⋯[aid]if i≥3
If i≥3 then n≥3 and, as i+1≤n, we have c1c2=U(P)a1a2so
h=U(P)f1=U(P)[dai+1]⋯a1a2⋯[aid],
a preconjugation of the cyclic permutation ai+1⋯aiof g.
If i=2 then Lemma 3.2.4.) applied to the four elements [a1c],[ca2],c
and d, shows that [cca2d]=[a2d]∈P. Therefore c1c2=U(P)[a1c][ca2d]=U(P)
26
a1[a2d]and
f1=U(P)[da3]⋯ana1[a2d],
as required.
If i=1 then from Lemma 3.2.5.) it follows that cd ∈Pso
f1=U(P)[(cd)a2]a3⋯an[a1(cd)],
as required.
If i=0 and n≥3 then the result follows, by symmetry, from the case
i=2, leaving the case i=0 and n=2. As gis cyclically reduced, a1a2a1a2
is a reduced word and therefore so is [a1c][ca2][a1c][ca2]. Hence [a1c][ca2]
is cyclically reduced. Applying Lemma 3.2.4.) to [ca2],d,[a1c]and c, gives
da1∈P. Similarly a2d∈Pand the result follows as before.
Finally, to show that 2.) implies 3.), suppose that h=U(P)f, where
f=[bg1]g2⋯gn−1[gnb],
and
g=gi+1⋯gng1⋯gi,
for some i. Then, from Lemma 4.1, fis cyclically reduced and hence, from
Theorem 3.4, his an interleaving of f. From Lemma 3.2, it follows that his
a cyclic interleaving of g.
Lemma 4.3. The system C†(Sε)is confluent.
Proof. The system Sεis standard and it is strongly confluent by Lemma 3.3.
Thus, by Theorem 2.5 it is enough to show that C†(Sε)is confluent on all
short cyclic words. Thus we have to consider the situation:
d∼
⊛
⇐Ô
C†(Sε)w∼
⊛
Ô⇒
C†(Sε)e∼,(5)
where wis short. We must show that
d∼
⊛
Ô⇒
C†(Sε)t∼
⊛
⇐Ô
C†(Sε)e∼,
for some t∼. As w∼is a short cyclic word we have w∼≤2. If w=1 in U(P),
then u∗
Ô⇒
Sε
1, for all u⊛
⇐⇒
C†(Sε)w, and we may take t∼=1. Thus, we may
assume 1 ≤w∼and w≠1∈U(P). If w∼=1 then w∼is cyclically reduced,
since w≠ε. Hence all rules involved in (5) are symmetric and we may take
t∼=w∼. Thus, from now on in the proof we may assume w∼=2. Since all
27
length preserving rules in ○
Ô⇒
C†(Sε)are symmetric, we are done if d∉Γ or e∉Γ.
Thus, as suggested by the notation we have d, e ∈Γ. Again, since length
preserving rules are symmetric, we may assume that the situation is
d∼
○
⇐Ô
C†(Sε)w0∼
○
⇐⇒
C†(Sε)⋯○
⇐⇒
C†(Sε)wk∼
○
Ô⇒
C†(Sε)e∼,
where all wihave length 2. As w0∼is not cyclically reduced, Lemma 4.1
implies that no wi∼is cyclically reduced. Hence, for all ithere exists ei∈Γ
such that wi∼
○
Ô⇒
C†(Sε)ei∈Γ. It therefore suffices to show that if
d∼
○
⇐Ô
C†(Sε)u∼
○
⇐⇒
C†(Sε)v∼
○
Ô⇒
C†(Sε)e∼,
where u=v=2 and d=e=1, then
d∼
⊛
⇐⇒
C1
e∼,
where C1is the length preserving part of C†(Sε). We may assume that
u∼=(ab)∼with a, b ∈Γ, [ab]=d∈Γ, and that there exists c∈Γ such that
either v∼=([ac][cb])∼or v=([ca][bc])∼. If v∼=([ac][cb])∼then
d∼
○
⇐Ô
C†(Sε)v∼
○
Ô⇒
C†(Sε)e∼,
so d∼
○
⇐⇒
C1
e∼and we are done.
Assume then that v∼=([ca][bc])∼. If ba ∈Pthen
d∼
○
⇐Ô
C†(Sε)u∼
○
Ô⇒
C†(Sε)[ba]∼,
and so d∼
○
⇐⇒
C1
[ba]∼. Also
e∼
○
⇐Ô
C†(Sε)v∼
○
Ô⇒
C†(Sε)[ba]∼,
so e∼
○
⇐⇒
C1
[ba]∼
○
⇐⇒
C1
d∼, as required.
Therefore we may assume that ba ∉P. Applying (P5) to the elements c,
a,band cwe have cab or abc ∈P. Also, from Lemma 3.2.3.), [bc][ca]∉P.
As v∼is not cyclically reduced it follows that [ca][bc]∈P, so we have
d∼
○
⇐Ô
C†(Sε)([cab]c)∼
○
Ô⇒
C†(Sε)[cabc]∼or d∼
○
⇐Ô
C†(Sε)(c[abc])∼
○
Ô⇒
C†(Sε)[cabc]∼,
28
and in both cases
d∼
○
⇐⇒
C1
[cabc]∼.
Moreover,
e∼
○
⇐Ô
C†(Sε)v∼
○
Ô⇒
C†(Sε)[cabc]∼,
so d∼
○
⇐⇒
C1
[cabc]∼
○
⇐⇒
C1
e, as required.
Having established the confluence of C†(Sε)we may get rid of the letter
εand the rule εÐ→ 1. That is: we switch back to the system S=S(P).
Theorem 4.4. Let S⊆Γ∗×Γ∗be the Thue system associated with P, c.f.
Defintion 3.1. Then C†(S)is geodesically perfect.
Proof. The system S=S(P)is a standard 2-monadic Thue system. The
confluence of C†(S)follows from Lemma 4.3. By Theorem 3.4 the semi-Thue
system Sis geodesically perfect. The result follows by Corollary 2.10.
Corollary 4.5. Cyclically reduced elements are minimal length representa-
tives of their conjugacy class in U(P). Let gand fbe cyclically reduced
elements of Γ∗such that gis conjugate to fin U(P). Then the following
hold.
1. gand fhave the same length.
2. If g∉P, i.e., g≥2, then we can transform the cyclic word g∼into the
cyclic word f∼by a sequence of at most glength preserving rules from
C†(S).
3. If g∈P, i.e., g=1, then we can transform ginto fby a sequence of
preconjugations.
Proof. Immediate by the confluence of C†(S)and Lemma 4.2.
The following theorem is the main result in this section. It makes state-
ment 2 of Corollary 4.5 much more precise.
Theorem 4.6. Let gand fbe a cyclically reduced elements of Γ∗such that
gis conjugate to fin U(P). Let g=g1⋯gnwith gi∈Pand n=g≥2. Then,
we may obtain f, as an element in U(P), by a single cyclic permutation
followed by a preconjugation. More precisely, we have
f=[bgi]⋯gng1⋯[gi−1b]∈U(P),
where b∈Pand bgi,gi−1b∈P.
29
Proof. This follows directly from Corollary 4.5.
We may strengthen the statement of Theorem 4.6 for pregroups which
satisfy certain extra conditions. First, in any pregroup Pwe can define a
canonical subgroup by
GP={x∈P (x, y),(y, x)∈D, ∀y∈P}.
We say that Psatisfies the extra axiom (P6) if the following is true.
(f, g )∉D∧(f, b)∈D∧(b, g)∈DÔ⇒ b∈GP.(P6)
Axiom (P6) holds for the standard pregroups defining amalgamated prod-
ucts or HNN-extensions (as in Section 3.1), but it does not hold in general
for the pregroup defining the fundamental group of a graph of groups, as
given in [24].
Remark 4.7. If Psatisfies the axiom (P6) then the element b∈Pin the
statement of Theorem 4.6 is necessarily in the canonical subgroup GP.
We say that Psatisfies the extra axiom (P7) if the following is true.
(y, z)∈D∧(x, [yz]) ∈D∧[yz]∉GP∧s∈{x, x}∧t∈{y , z}
Ô⇒ {(s, t),(t, s)} ⊂D. (P7)
First note that (P7) implies axiom (P6). To see this, suppose that b, f and
gsatisfy the hypotheses of (P6). Let z=g,y=[bg]and x=[f b]. Then
(y, z)∈D,[yz]=band (x, [yz]) =(x, b)=([fb], b)∈D. If [yz]∉GPthen
(P7) gives (x, y)∈Dand this implies that (f b, bg)∈D, from which we infer
(f, g )∈D, a contradiction. Thus we must have [yz]∈GPand (P6) holds.
Axiom (P7) holds for the standard pregroup defining an amalgamated
product, but not, in general, for the standard pregroup defining an HNN-
extension (as in Section 3.1). In contrast the following axiom (P8) holds for
the standard pregroup of an HNN-extension, but not for that of an amalga-
mated product.
(a, b)∈D∧[ab]=cÔ⇒ a∈GP∨b∈GP∨c∈GP.(P8)
Again, axiom (P8) implies axiom (P6). Indeed, consider the condition (b, g)∈
D. If we have [bg]∈GP, then (f, b)∈Dimplies (f , g)∈D, contrary to the
hypothesis of (P6). Given (f, g )∉D, we can exclude g∈GP. Thus, (P8)
yields the implication of (P6).
In pregroups in which axiom (P7) holds, elements of Pbehave well with
respect to preconjugation.
30
Lemma 4.8. Let Pbe a pregroup satisfying axiom (P7), let H=GPbe its
canonical subgroup and let a, b ∈P. If c∈Pis a preconjugate of both aand
bthen either c∈Hor bis a preconjugate of a.
Proof. Let c=[uau]=[vbv], for some u, v ∈P. Then either ua ∈Por au ∈P.
Assume ua ∈P. We have b=[vcv]∈P, so either vc ∈Por cv ∈P. Assume
c=[[ua]u]∉H. Then vc ∈Ptogether with (P7) implies uv ∈P. Similarly
cv ∈Pimplies uv ∈P. By symmetry, if au ∈Pand c∉Hthen again uv ∈P.
Therefore, either c∈Hor b=[(vu)a(uv)], a preconjugate of a.
In pregroups in which axiom (P8) holds, elements of P∖GPbehave well
with respect to preconjugation.
Lemma 4.9. Let Pbe a pregroup satisfying axiom (P8) and H=GPits
canonical subgroup. Let a∈P∖Hand b∈P.
1.) If bis a preconjugate of athen b=[hah], for some element h∈H.
2.) If bis conjugate to athen bis a preconjugate of a.
Proof. 1. If b=[c ac], where c∈P, then either ca ∈Por ac ∈P. By
symmetry, assume ca ∈P. If c∉Hthen (P8) implies [ca]=h∈H, so
c=ah. Thus b=[cac]=[hah], as required.
2. From Corollary 4.5.3 there exist a sequence of elements a=b0,...,bn=
b, of P, such that bi+1is a preconjugation of bi, for all i. As each
preconjugation is by an element of Hit follows that bis in fact a
preconjugate of a.
5 The conjugacy problem in amalgamated prod-
ucts and HNN-extensions
5.1 Conjugacy in amalgamated products
As in Section 3.1, the defining pregroup for the group G=A∗HBcan be
chosen to be P=A∪B; and the common subgroup His then equal to the
canonical subgroup GP. Therefore Psatisfies (P6). Conjugacy of elements
of a free product with amalgamation is described in [21], which now follows
easily from of Corollary 4.5 and Theorem 4.6 as we show below. First we
state the theorem.
31
Theorem 5.1 ([21], Thm. 4.6).Let G=A∗HB. Every element of Gis
conjugate to a cyclically reduced element of G. (That is an element gwhich
can be written as g=g1⋯gnwith gi∈A∪Band either n=1or gi−1and gi
do not lie in the same factor for all i∈ZnZ.) If gis a cyclically reduced
element of Gthen the following hold.
1. If gis conjugate to h∈Hthen g∈A∪Band there exists a sequence
h, h1,...,hℓ, g where hi∈Hand consecutive terms are conjugate in
some factor.
2. If 1 does not hold and gis conjugate to an element f∈A∪B, then
gand fbelong to the same factor, Aor B, and they are conjugate in
that factor.
3. If n=g≥2, then 1 and 2 do not hold. If gis conjugate to a cyclically
reduced element f, then fcan be written as f=h−1gi⋯gng1⋯gi−1h, for
some h∈Hand iwith 1≤i≤n.
Proof. Assertion 3 is a trivial consequence of Theorem 4.6. Indeed, for n≥2
Theorem 4.6 says that f=b−1gi⋯gng1⋯gi−1bwhere b, b−1gi, gi−1b∈A∪B.
However, Psatisfies (P6), hence b∈Hby Remark 4.7. Moreover, 1 or 2
implies n=1 by Corollary 4.5, 1.
Thus, let n=1 and g, p ∈A∪Bbe conjugate to each other. Applying
Corollary 4.5, 3, there is a sequence p=p0, p1,...,pℓ=gwhere consecutive
terms are preconjugate, i.e., consecutive terms are conjugate in some factor.
From Lemma 4.8, either every piis in Hor the sequence may be shortened.
Thus, if gis not conjugate to any h∈H, we may assume g∈A∖Hand l=1,
so pis a preconjugate of g; that is of the form a−1ga for some a∈A. Hence
p=a−1ga ∈A∖H, giving 2.
Otherwise every piis in Hand 1 holds.
5.2 Conjugacy in HNN-extensions
As in Section 3.1, for G=HNN(H, t;t−1At =B)the defining pregroup can
be chosen as P=H∪Ht−1H∪HtH; and the base group His then equal to
the canonical subgroup GP. Therefore Psatisfies (P8).
The word problem in Gcan be solved, if we can effectively perform Britton
reductions, see e.g. in [20]: we read non-trivial elements in Gas words over
H∖{1}and in t±1. Whenever we see a factor in t−1At, then we replace it by
the corresponding factor in B. Similarly, whenever we see a factor in tBt−1,
then we replace it by the corresponding factor in A. This leads to a normal
32
form where each gbecomes an element in H: that is, for some uniquely
defined t-sequence of minimal length n≥1, the element ghas the form
g=h0tε1h1⋯tεn−1hn−1tεnhn.
In order to perform a cyclic reduction we remove h0tε1h1from the left and
put it at the right. We continue with Britton and cyclic reductions for as long
as possible and eventually reach a (Britton) cyclically reduced form. Clearly
every cyclically reduced form g, with non-trivial t-sequence, is conjugate to
and element of the form
tε1z1⋯tεn−1zn−1tεnzn,(6)
where tε1,...,tεnis the t-sequence of g,zi∈Hand, for all iwe have
tεizitεi+1∉t−1At ∪tBt−1,
(subscripts modulo n). Cyclically reduced elements which either belong to
Hor are written in the form of (6) are called standard cyclically reduced
elements of G. In terms of the pregroup P, every pregroup cyclically reduced
word can be written as a preconjugate, by an element of H, of a standard
cyclically reduced word; and conversely, every standard cyclically reduced
word is cyclically reduced with respect to P.
The conjugacy theorem for HNN-extensions, Collins’ Lemma, can be
found in [20, Chapter IV, Theorem 2.5], and is stated for standard cycli-
cally reduced words. In analogy to amalgamated products we restate it as
follows.
Theorem 5.2 (D.J. Collins (1969)).Let G=HNN(H, t;t−1At =B)be an
HNN-extension over two isomorphic subgroups A, B in some base group H.
Every element of Gis conjugate to a standard cyclically reduced element. Let
gand fbe conjugate, standard cyclically reduced elements of G. Then the
following (mutually exclusive) statements hold.
1. If f∈A∪Bthen there exists a sequence f=c0, c1,...,cℓ=gof elements
of A∪B, such that, for i=1,...,ℓ, we have ci=k−1
it−δici−1tδiki, with
ki∈H,δi=±1and t−δici−1tδi∈t−1At ∪tBt−1.
2. If gis not conjugate to an element of A∪Band f∈Hthen gand f
are conjugate by an element of the base group H.
3. If fis not in Hthen there exist n≥1,zi∈H,εi=±1,1≤j≤nand
c∈A∪B, such that g=tε1z1⋯tεnznand fhas t-sequence of length n
and is equal in Gto
c−1tεjzj⋯tεnzntε1z1⋯tεj−1zj−1c,
with c∈A, if εj=−1; and c∈B, if εj=1.
33
Proof. As in the proof of Theorem 5.1, it follows from Theorem 4.6 that if g=
tε1z1⋯tεnzn, where n≥2, then fis equal in Gto c−1tεjzj⋯tεnzntε1z1⋯tεj−1zj−1c,
for some c∈P, and as (P6) holds we have c∈H.
The pregroup Pfor Gsatisfies (P8) and His the canonical subgroup.
Therefore P∖H=HtH ∪Ht−1H. Hence, if f=tεz, for some z∈Hand
ε=±1, then from Corollary 4.5 and Lemma 4.9 every cyclically reduced
conjugate of fhas the form c−1tεzc, for some c∈H. Since gand fare
standard, statement 3 holds in both these cases.
This leaves the case where f∈H. As in the proof of Theorem 5.1, applying
Corollary 4.5, 3, there is a sequence f=p0, p1,...,pℓ=gof elements of P,
where consecutive terms are preconjugate, say pi=q−1
ipi−1qi, with qi∈P. If
pi∈P∖Hthen, from Lemma 4.9, the sequence may be shortened, at least
while ℓ>1. Then, since pℓ−1∈Hwe have, from (P8), g∈H. Hence we may
assume that either pi∈H, for i=0,...,ℓ. Again, if qi∈Hand ℓ>1 then
the sequence may be shortened, so we may assume that either qi∈P∖H, for
i=1,...,ℓ; or that ℓ=1.
Applying (P8), q−1
ipi−1qi=pi∈H, with qi∉H, implies that piis conjugate
to an element of A∪B. If gis not conjugate to an element of A∪Bit follows
that ℓ=1, and q1∈H, as required in statement 2. Otherwise 1 holds.
6 The conjugacy problem in virtually free groups
We consider only the case of finitely generated virtually free groups. Virtually
free groups are hyperbolic, and it has been shown by Epstein and Holt [8]
that the conjugacy problem for hyperbolic groups can be solved in linear
time. Hence, the following is a special case of [8]. However, our algorithm is
much simpler and more direct. It can be implemented in a straightforward
way using finite pregroups.
Proposition 6.1. The conjugacy problem in finitely generated virtually free
groups can be solved in linear time.
Proof. A finitely generated virtually free group Gis the universal group U(P)
of some finite pregroup P, see Propostion 3.6. As above let Γ =P∖{ε}.
By a standard procedure involving Theorem 3.4 we can compute cyclically
reduced elements in linear time. Thus, we may assume that our input words
are given as g=g1⋯gnand f=f1⋯fnwith gi,fi∈Γ such that both sequences
are cyclically reduced. For n=1 we can use table look-up. Hence we may
assume n≥2 henceforth.
Now, let us put a linear order on Γ. Then the shortlex normal form of g
begins with a letter [g1a1]such that the geodesic length of a1g2⋯gnis n−1.
34
But this implies a1g2∈P. Thus, working from left to right we may compute
the shortlex normal form of g, in linear time; and we may assume that this
is g1⋯gn
We know f=[bgi]⋯gng1⋯[gi−1b]∈U(P)by Theorem 4.6. Thus, bf b =
gi⋯gng1⋯gi−1and the number of all bfb is bounded by a constant dependent
only on the order of P. Hence, we may assume that f=gi⋯gng1⋯gi−1. Since
the word problem in finitely generated virtually free groups can be solved in
linear time (e.g., using the system S(P)or by computing the shortlex normal
form), we may assume 2 <i<n. (We also see that the conjugacy problem
can be solved in quadratic time: but our goal is linear time.)
Now, the shortlex normal form of g2can be written as
g1⋯gn−1[gna1][a1g1a2]⋯[angn],
for appropriate ai∈P. As a consequence,
fai=gi⋯gn−1[gna1][a1g1a2]⋯[ai−1gi−1ai].
However, the word gi⋯gn−1[gna1][a1g1a2]⋯[ai−1gi−1ai]is in shortlex normal
form. Therefore the shortlex normal form of fis f′[ai−1gi−1], where
f′=gi⋯gn−1[gna1][a1g1a2]⋯[ai−2gi−2ai−1].
Thus, it is enough to compute the shortlex normal form ̂
f=f′p, of f.
Erasing the last letter pyields f′. We can run the pattern matching algorithm
of Knuth-Morris-Pratt, in linear time, in order to obtain a list (i1,...,ik)with
2<ij<nwhere the pattern f′appears as gij⋯gn−1[gna1][a1g1a2]⋯[aij−2gij−2aij−1].
All that remains is to verify whether or not [paij]=[aij−1gij−1aij], for one
index in the list.
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