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Trickle groups
PAOLO BELLINGERI
EDDY GODELLE
LUIS PARIS
Abstract A new family of groups, called trickle groups, is presented. These groups generalize
right-angled Artin and Coxeter groups, as well as cactus groups. A trickle group is defined by a
presentation with relations of the form xy =zx and xµ=1, that are governed by a simplicial graph,
called a trickle graph, endowed with a partial ordering on the vertices, a vertex labeling, and an
automorphism of the star of each vertex. We show several examples of trickle groups, including
extended cactus groups, certain finite-index subgroups of virtual cactus groups, Thompson group F,
and ordered quandle groups. A terminating and confluent rewriting system is established for trickle
groups, enabling the definition of normal forms and a solution to the word problem. An alternative
solution to the word problem is also presented, offering a simpler formulation akin to Tits’ approach
for Coxeter groups and Green’s for graph products of cyclic groups. A natural notion of a parabolic
subgraph of a trickle graph is introduced. The subgroup generated by the vertices of such a subgraph
is called a standard parabolic subgroup and it is shown to be the trickle group associated with the
subgraph itself. The intersection of two standard parabolic subgroups is also proven to be a standard
parabolic subgroup. If only relations of the form xy =zx are retained in the definition of a trickle
group, then the resulting group is called a preGarside trickle group. Such a group is proved to be a
preGarside group, a torsion-free group, and a Garside group if and only if its associated trickle graph
is finite and complete.
AMS Subject Classification Primary: 20F10, Secondary: 20F05, 20F36, 20F55, 20F65.
Keywords Trickle groups, right-angled Coxeter groups, right-angled Artin groups, cactus groups,
virtual cactus groups, Thompson group F, word problem, rewriting systems, preGarside groups,
Garside groups.
1 Introduction
There are numerous groups in the literature defined by relations of the form xy =zx, often with
additional constraints on the orders of generators. Prominent examples include right-angled Artin
groups, right-angled Coxeter groups, and more generally, graph products of cyclic groups. The aim of
the present paper is to study a specific family of such groups that we call trickle groups.
Cactus groups are emblematic examples of trickle groups. These groups first appeared as quasi-
braid groups in the study of the mosaic operad [Dev99,EHK+10,KW19] and they were subsequently
generalized to all Coxeter groups [DJS03]. Their significance was further highlighted in their connection
to coboundary categories [HK06], mirroring the role of braid groups in braided categories. Note that the
term “cactus groups” was coined in [HK06]. Additionally, cactus groups and their generalizations to
Coxeter groups have found applications in representation theory under various guises [KTW04,Bon16,
Los19,CGP20,RW24].
arXiv:2412.04932v1 [math.GR] 6 Dec 2024
2P Bellingeri, E Godelle and L Paris
For n∈N≥2, the cactus group Jnis defined by the presentation with generators xp,q, 1 ≤p<q≤n,
and relations:
(j1) x2
p,q=1, for 1 ≤p<q≤n,
(j2) xp,qxm,r=xm,rxp,q, for [p,q]∩[m,r]=∅,
(j3) xp,qxm,r=xp+q−r,p+q−mxp,q, for [m,r]⊂[p,q].
The elements of Jnare often depicted using planar diagrams. More precisely, an element g∈Jnis
represented by an n-tuple of smooth paths in the plane, b=(b1,...,bn), bi: [0,1] →R2, satisfying
the following conditions.
• There exists a permutation σ∈Snsuch that bi(0) =(0,i) and bi(1) =(1, σ(i)), for all
i∈ {1,...,n}.
• For all t∈[0,1] and all i∈ {1,...,n}we have π1(bi(t)) =t, where π1:R2→Rdenotes the
projection onto the first coordinate.
• Crossings between the bi’s may be multiple but are always transversal.
The generator xp,qis represented in Figure 1.1 and relation (j3) is illustrated in Figure 1.2.
p
q
Figure 1.1: Generator of Jn
=
Figure 1.2: Relation (j3) in the presentation of Jn
The extension of this definition to Coxeter groups is straightforward. Let (W,S) be a Coxeter system
associated with a Coxeter graph Υ. For X⊆S, the subgroup of Wgenerated by Xis called a standard
parabolic subgroup and is denoted by WX, and the full Coxeter subgraph of Υspanned by Xis denoted
by ΥX. We say that X⊆Sis irreducible if ΥXis connected, and we say that Xis of spherical type
if WXis finite. In the latter case WXcontains a unique element of maximal length (with respect to S),
denoted by wX, and this element satisfies wXXw−1
X=Xand w2
X=id (see [Bou68]). We denote the set
of non-empty irreducible and spherical subsets of Sby Sfi.
The cactus group C(W,S) associated with (W,S) is defined by the presentation with generators xX,
X∈ Sfi, and relations:
Trickle groups 3
(j1) x2
X=1, for X∈ Sfi,
(j2) xXxY=xYxX, if ΥX∪Yis the disjoint union of ΥXand ΥY, meaning that X∩Y=∅and st =ts
for all s∈Xand t∈Y,
(j3) xXxY=xwX(Y)xX, for Y⊂X.
Our aim is to investigate the combinatorial properties of these groups within a broader framework that
encompasses various more or less natural generalizations of cactus groups.
Let Γbe a simplicial graph. We denote the vertex set of Γby V(Γ) and the edge set by E(Γ).
The link of a vertex x∈V(Γ) , denoted by linkx(Γ), is the full subgraph of Γspanned by {y∈
V(Γ)| {x,y} ∈ E(Γ)}. The star of x, denoted by starx(Γ), is the full subgraph of Γspanned by
{y∈V(Γ)| {x,y} ∈ E(Γ)}∪{x}.
Atrickle graph is a quadruple (Γ,≤, µ, (ϕx)x∈V(Γ)), where:
•Γis a simplicial graph,
•≤is a (partial) order on V(Γ),
•µ:V(Γ)→N≥2∪ {∞} is a vertex labeling,
•ϕx: starx(Γ)→starx(Γ) is an automorphism of starx(Γ) for all x∈V(Γ),
that must satisfy certain conditions defined in Section 2.
The trickle group Tr(Γ) associated with a trickle graph Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)) is the group defined
by the following presentation.
Tr(Γ)=⟨V(Γ)|xµ(x)=1 for all x∈V(Γ) such that µ(x)=∞, ϕx(y)x=ϕy(x)y
for all {x,y} ∈ E(Γ)⟩.
One of the conditions in the definition of a trickle graph in Section 2entails that, for any edge
{x,y} ∈ E(Γ), either ϕx(y)=yor ϕy(x)=x. Consequently, the above presentation is indeed a
presentation with relations of the form xy =zx and xµ=1 . Another immediate consequence of these
conditions is that the trivial order is admissible, and therefore right-angled Coxeter groups, right-angled
Artin groups and, more generally, graph products of cyclic groups are trickle groups.
As mentioned above, cactus groups are trickle groups, where here µ(x)=2 for all x∈V(Γ). Trickle
groups include other groups naturally related to cactus groups, such as the “Artin” versions of cactus
groups (associated with Coxeter groups), where we keep relations (j2) and (j3) and ignore relations
(j1). In this case, such a group is also a preGarside group in the sense of [GP13] (see Section 7and
Subsection 2.5). More generally, the example of cactus groups associated with Coxeter systems can
be naturally extended to families of subgroups of a given group Gsatisfying certain properties that are
presented in Subsection 3.1.
Our study is also an opportunity to investigate virtual cactus groups since, as we will see in Subsection
3.2, they contain finite index subgroups that are trickle groups. Let S1,...,Sℓbe a collection of
circles immersed in the plane having only double transverse crossings. We assign to each crossing a
“positive”, “negative” or “virtual” value that we indicate on the graphical representation of S1∪ · · · ∪ Sℓ
as in Figure 1.3. Such a figure is called a virtual link diagram. We consider the equivalence relation on
4P Bellingeri, E Godelle and L Paris
positive negative virtual
Figure 1.3: Crossings in a virtual link diagram
the set of virtual link diagrams generated by the isotopy and the virtual Reidemeister moves as defined
by Kauffman [Kau99,Kau00]. An equivalence class of virtual link diagrams is a virtual link.
Since the publication of Kauffman’s seminal paper [Kau99], the theory of virtual knots and links has
grown significantly, and this notion has been extended to other combinatorial and/or topological objects
represented by planar diagrams. The leitmotif underlying these theories is that two arcs connecting the
same points and passing only through virtual crossings are equivalent.
Since the elements of the cactus group Jnare represented by planar diagrams, it is natural to extend Jn
by adding virtual crossings to the cactus crossings while keeping the principle that two arcs connecting
the same points and passing only through virtual crossings are equivalent. Then we obtain the virtual
cactus group, VJn, which will be studied in detail in Subsection 3.2. Note that these groups are not new,
having been introduced in [IKL+23], where it is shown that VJnis the Sn-equivariant fundamental
group of the real form of the “cactus flower moduli space” ¯
Fn.
The connection between virtual cactus groups and trickle groups mirrors that between virtual braid
groups and Artin groups [GP12a,BCP16,BP20,BPT23]. We prove that VJncan be decomposed as
a semi-direct product VJn=KJn⋊Sn, where Snis the symmetric group on {1,...,n}, and KJnis
a trickle group (see Proposition 3.7). This decomposition enables us to address the word problem in
VJn, to define normal forms for the elements of VJn, and to show that Jnembeds into VJn.
It is probable that numerous other trickle groups exist within the literature. Among these, we have
pinpointed two specific examples: one originating from dynamical systems and the other from knot
theory.
Thompson group F, introduced by Richard Thompson in 1965 in an unpublished manuscript, is a group
of homeomorphisms of the real line with many unusual properties. For instance, its derived group F′
is a simple group, the quotient F/F′is a free abelian group of rank 2, and it contains no subgroup
isomorphic to the rank 2 free group. We refer to [CFP96] for a general overview on this group. A
well-known presentation for Fis as follows:
⟨xn,n∈N|xkxn=xn+1xkfor k<n⟩,
which, while featuring relations of the form xy =zx, does not fulfill all the requirements of a trickle
group presentation. However, in Subsection 3.3 we show another presentation for Fthat is indeed a
trickle presentation (see Theorem 3.15). So, Thompson group Fis a trickle group. This structure is
particularly noteworthy as any “natural standard parabolic subgroup” of Fis a copy of Fitself (see
Proposition 3.16). Moreover, Theorem 2.14 will imply that Fis a preGarside group.
Quandles are algebraic structures whose axioms reproduce the Reidemeister moves in knot theory.
Independently introduced by Joyce [Joy99] and Matveev [Mat82], they have been frequently used to
construct knot or link invariants. As noted by Joyce [Joy99] and Matveev [Mat82], a classification of
Trickle groups 5
quandles would de facto entail a classification of knots. This explains the difficulty of studying the set
of all quandles, leading researchers to focus on specific families of quandles. Ordered quandles were
introduced recently in this perspective [BPS22,DDH+07], and it turns out that they are a disguised
form of trickle graphs. The details of this construction are given in Subsection 3.4.
As previously mentioned, the objective of this paper is a combinatorial study of trickle groups. In
Section 4we determine a confluent and terminating rewriting system for these groups (see Theorem
2.4). According to Newman [New42], this enables the definition of normal forms for their elements
and, consequently, a solution to the word problem (see Corollary 2.5). Moreover, it yields other
immediate results, such as a characterization of finite trickle groups (see Corollary 2.7). Notice that
our rewriting system and its associated normal forms are not novel for graph products of cyclic groups
[Wyk94,HM95,CGW09], and a similar rewriting system for classical cactus groups was considered
in [Gen22]. Note also that the term “trickle”, used to designate trickle groups, originates from this
algorithm because it metaphorically involves pushing down as many syllables as possible using only
relations of the form xy =zx .
The remaining sections of the paper explore how trickle groups behave in a manner analogous to groups
studied in the theory of Coxeter, Artin, and Garside groups. Building upon the algorithms and methods
introduced in Section 4, we present in Section 5an alternative algorithm for solving the word problem
in a trickle group. This new algorithm offers a simpler formulation compared to the one presented in
Section 4. Furthermore, it aligns more closely with the algorithms described in [Tit69] for Coxeter
groups, in [Gre90] for graph products of cyclic groups, and more generally, in [PS23] for Dyer groups.
A natural notion of a parabolic subgraph emerges in the context of trickle graphs. This leads to the
natural question of studying the trickle groups defined by such subgraphs. Drawing a parallel with the
theory of Coxeter and Artin groups, we refer to these as standard parabolic subgroups. As the name
suggests, we establish in Section 6that a standard parabolic subgroup is indeed a subgroup of the trickle
group associated with the original graph (see Theorem 2.10). Moreover, we prove that the intersection
of two standard parabolic subgroups is a standard parabolic subgroup (see Corollary 2.12).
Section 7establishes a connection between trickle groups and Garside theory. A preGarside trickle
graph is defined as a trickle graph Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)) for which µ(x)=∞for all x∈V(Γ).
ApreGarside trickle group is a trickle group associated with a preGarside trickle graph. Additionally,
one can define an associated preGarside trickle monoid using the same presentation, but interpreted as
a monoid presentation. The term “preGarside” originates from [GP13], where the authors investigate
monoids and groups that they call preGarside monoids and preGarside groups. Notable examples of
such monoids and groups include all Artin monoids and all Artin groups. Garside monoids and Garside
groups are also preGarside monoids and preGarside groups. In Section 7, we prove that a preGarside
trickle monoid is indeed a preGarside monoid and that a preGarside trickle group is a preGarside group
(see Theorem 2.14). Furthermore, the monoid and the group are respectively a Garside monoid and a
Garside group if and only if Γis finite and complete (see Theorem 2.15). This introduces new examples
of Garside groups to which we can associate Coxeter-style quotients.
Another objective of Section 7is to address certain questions that arise for preGarside monoids and
groups within the context of preGarside trickle monoids and groups. First, we already know from
Section 4that a preGarisde trickle group has a solution to the word problem. Then, we prove in Section
6P Bellingeri, E Godelle and L Paris
7that, if the vertex set of the graph is finite, then the preGarside trickle group is torsion-free (see
Theorem 2.16). The remaining questions concern the relationship between monoids and groups. These
inquiries, posed in [GP13], are specifically focused on preGarside monoids and groups. We prove that
a preGarside trickle monoid embeds into its enveloping group (see Theorem 2.17) and we prove several
results concerning its parabolic submonoids and subgroups (see Theorem 2.18).
Trickle groups appear to be a reasonable generalization of graph products of cyclic groups, and con-
sequently, of right-angled Artin groups and right-angled Coxeter groups. Therefore, it is natural to
explore whether results known for some or all graph products of cyclic groups can be extended to some
or all trickle groups. Relevant questions in this direction include:
(1) Do the normal forms described in Section 4form a regular language? Are trickle groups automatic
or bi-automatic?
(2) Which trickle groups admit geometric actions on CAT(0) cube complexes?
(3) Are trickle groups residually finite? Are preGarside trickle groups residually nilpotent without
torsion? Can we determine the Lie algebra associated with their lower central series, as done for
right-angled Artin groups (see [DK92])?
(4) Which preGarside trickle groups are orderable, bi-orderable, or admit isolated orders?
Additionally, it would be valuable to discover new examples of trickle groups, particularly those arising
from areas of mathematics beyond group theory.
The paper is organized as follows. Section 2presents the fundamental definitions and precise statements
of our main results. It is divided into five subsections. In the first subsection we introduce the concepts
of trickle graphs and trickle groups, along with illustrative examples like graph products of cyclic
groups. In Subsection 2.2 we state the main results of Section 4, which focuses on the trickle algorithm.
In Subsection 2.3 we state the main results of Section 5, which focuses on the Tits-style algorithm.
In Subsection 2.4 we state the main results of Section 6, which focuses on parabolic subgroups.
In Subsection 2.5 we state the main results of Section 7, which focuses on preGarside trickle groups.
Section 3is devoted to examples and it is divided into four subsections: Subsection 3.1 for cactus groups
in their generalized version, Subsection 3.2 for virtual cactus groups, Subsection 3.3 for Thompson
group F, and Subsection 3.4 for ordered quandle groups. As mentioned earlier, Sections 4,5,6and
7contain the proofs: those concerning the trickle algorithm in Section 4, those concerning the Tits-
style algorithm in Section 5, those concerning parabolic subgroups in Section 6, and those concerning
preGarside trickle groups in Section 7.
Acknowledgments This work originated during a research residency program titled “Cactus and
Posets” at CIRM (Luminy, Marseille, France) from November 7th to 11th, 2022. The three authors
extend their sincere gratitude to CIRM for the generous support and resources provided (funding,
dedicated workspaces, library access, etc.), without which this project would not have been possible.
Trickle groups 7
2 Definitions and statements
2.1 Definitions and first examples
The set of vertices of a simplicial graph Γis denoted by V(Γ) and the set of its edges is denoted
by E(Γ). The link of a vertex x∈V(Γ), denoted by linkx(Γ), is the full subgraph of Γspanned by
{y∈V(Γ)| {x,y} ∈ E(Γ)}, and the star of x, denoted by starx(Γ), is the full subgraph of Γspanned
by {y∈V(Γ)| {x,y} ∈ E(Γ)}∪{x}.
Definition Atrickle graph is a quadruple (Γ,≤, µ, (ϕx)x∈V(Γ)), where
•Γis a simplicial graph,
•≤is a (partial) order on V(Γ),
•µis a labeling µ:V(Γ)→N≥2∪ {∞} of the vertices,
•ϕx: starx(Γ)→starx(Γ) is an automorphism of starx(Γ) for all x∈V(Γ).
For x,y∈V(Γ) the notation x||ymeans that xand yare not comparable in the sense that x≤ yand
y≤ x. We set E||(Γ)={{x,y} ∈ E(Γ)|x||y}. The quadruple (Γ,≤, µ, (ϕx)x∈V(Γ)) must satisfy the
following conditions.
(a) For all x,y∈V(Γ), if x<y, then {x,y} ∈ E(Γ) .
(b) For all x,y,z∈V(Γ), if {x,y} ∈ E||(Γ) and z≤y, then {x,z} ∈ E||(Γ).
(c) For all x∈V(Γ) and all y,z∈starx(Γ), we have z≤yif and only if ϕx(z)≤ϕx(y).
(d) For all x∈V(Γ) and all y∈starx(Γ), if ϕx(y)=y, then y<x.
(e) For all x∈V(Γ), if µ(x) is finite, then ϕxhas finite order and its order divides µ(x).
(f) For all x∈V(Γ) and all y∈starx(Γ), µ(ϕx(y)) =µ(y).
(g) For all x,y,z∈V(Γ), if z<y<x, then (ϕx◦ϕy)(z)=(ϕy′◦ϕx)(z) , where y′=ϕx(y).
We will often say that Γis a trickle graph meaning that, implicitly, ≤,µand (ϕx)x∈V(Γ)are also given.
Remark Let x,y,z∈V(Γ) be such that z≤y≤x. Then it is easily seen that (ϕx◦ϕy)(z)=
(ϕy′◦ϕx)(z), where y′=ϕx(y), if either z=yor y=x. So, Condition (g) in the definition of a trickle
graph also holds if at least two of the three vertices are equal.
Definition The trickle group Tr(Γ) associated with a trickle graph Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)) is the
group defined by the following presentation.
Tr(Γ)=⟨V(Γ)|xµ(x)=1 for x∈V(Γ) such that µ(x)=∞, ϕx(y)x=ϕy(x)yfor {x,y} ∈ E(Γ)⟩.
Remark By Condition (d) in the definition of a trickle graph, if {x,y} ∈ E||(Γ) , then the relation
ϕx(y)x=ϕy(x)ybecomes yx =xy. If x<y, then this relation becomes yx =ϕy(x)y. So, Tr(Γ) has
a presentation with relations of the form xy =zx and xµ=1.
8P Bellingeri, E Godelle and L Paris
Example 1 Let Γbe a simplicial graph and let µ:V(Γ)→N≥2∪ {∞} be a labeling of the vertices.
To the pair (Γ, µ) we associate the graph product of cyclic groups G(Γ, µ) defined by the following
presentation.
G(Γ, µ)=⟨V(Γ)|xµ(x)=1 for x∈V(Γ) such that µ(x)=∞,xy =yx for {x,y} ∈ E(Γ)⟩.
If µ(x)=∞for all x∈V(Γ), then G(Γ, µ) is a right-angled Artin group, and if µ(x)=2 for all
x∈V(Γ), then G(Γ, µ) is a right-angled Coxeter group. It is easily seen that G(Γ, µ) is a trickle group,
where ≤is the trivial order defined by x≤yif and only if x=y, and ϕxis the identity of starx(Γ) for
all x∈V(Γ).
Example 2 Le Υbe a Coxeter graph and let (W,S) be its associated Coxeter system. As in the
introduction, for X⊆S, we denote by WXthe standard parabolic subgroup generated by Xand by
ΥXthe full Coxeter subgraph of Υspanned by X. Recall that X⊆Sis called irreducible and of
spherical type if ΥXis connected and WXis finite. Recall also that, in this case, WXcontains a unique
element of maximal length (with respect to S), denoted by wX, and this element satisfies w2
X=1 and
wXXwX=X(see [Bou68]). We denote by Sfithe set of non-empty subsets X⊆Sthat are irreducible
and of spherical type.
We define Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)) as follows. The set of vertices of Γis a set V(Γ)={xX|X∈ Sfi}
in one-to-one correspondence with Sfi. Two vertices xXand xYare connected by an edge if either
X⊂Y, or Y⊂X, or ΥX∪Yis the disjoint union of ΥXand ΥYin the sense that X∩Y=∅and
{s,t} ∈ E(Υ) for all s∈Xand t∈Y. We set xX≤xYif X⊆Y. We set µ(xX)=2 for all X∈ Sfi. Let
xX∈V(Γ) and xY∈V(starxX(Γ)). We set ϕxX(xY)=xwX(Y)if Y⊂Xand ϕxX(xY)=xYotherwise. It is
easily verified that ≤is a (partial) order on V(Γ) and that, for all xX∈V(Γ), ϕxXis an automorphism
of starxX(Γ).
Now, we prove that Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)) satisfies Conditions (a) to (g) of the definition of a trickle
graph.
Lemma 2.1 The quadruple Γ = (Γ,≤, µ, (ϕx)x∈V(Γ))above defined is a trickle graph.
Proof Conditions (a), (d) and (f) are satisfied by definition of Γ.
We show that Γsatisfies Condition (b). Let X,Y,Z∈ Sfibe such that {xX,xY} ∈ E||(Γ) and xZ≤xY.
Then ΥX∪Yis the disjoint union of ΥXand ΥYand Z⊂Y. Thus, ΥX∪Zis the disjoint union of ΥX
and ΥZ, hence {xX,xZ} ∈ E||(Γ).
We show that Γsatisfies Condition (e). Let X∈ Sfi. Since wXhas order 2 , we have ϕ2
xX(xY)=
xw2
X(Y)=xYif Y⊂X. We have ϕ2
xX(xY)=ϕxX(xY)=xYfor every other vertex of starxX(Γ), hence the
order of ϕxXdivides µ(xX)=2.
We show that Γsatisfies Condition (c). Let X,Y,Z∈ Sfibe such that xZ,xY∈starxX(Γ) . Since, by
Condition (e) already proved, ϕ2
xX=id, to show the equivalence xZ≤xY⇔ϕxX(xZ)≤ϕxX(xY),
it suffices to show the implication xZ≤xY⇒ϕxX(xZ)≤ϕxX(xY). Suppose xZ≤xY, that is,
Z⊆Y. If ΥX∪Yis the disjoint union of ΥXand ΥY, then ΥX∪Zis the disjoint union of ΥX
and ΥZ, hence ϕxX(xZ)=xZ≤xY=ϕxX(xY). If xX≤xZ≤xY, then X⊆Z⊆Y, hence
Trickle groups 9
ϕxX(xZ)=xZ≤xY=ϕxX(xY). If xZ≤xX≤xY, then Z⊆X⊆Yhence wX(Z)⊆X⊆Y, and
therefore ϕxX(xZ)=xwX(Z)≤xX≤xY=ϕxX(xY). If xX≤xYand xZ||xX, then X⊆Y,Z⊆Yand
ΥX∪Zis the disjoint union of ΥXand ΥZ, hence ϕxX(xZ)=xZ≤ϕxX(xY)=xY. If xZ≤xY≤xX, then
Z⊆Y⊆X, hence wX(Z)⊆wX(Y)⊆X, and therefore ϕxX(xZ)=xwX(Z)≤xwX(Y)=ϕxX(xY).
Finally we show that Γsatisfies Condition (g). Let X,Y,Z∈ Sfibe such that Z⊆Y⊆X. Let
Y′=wX(Y). We have
(wXwY)(Z)=(wXwYw−1
XwX)(Z)=(wwX(Y)wX)(Z)=(wY′wX)(Z),
hence (ϕxX◦ϕxY)(xZ)=(ϕxY′◦ϕxX)(xZ).
It is obvious that the cactus group C(W,S) is equal to the trickle group Tr(Γ).
Example 3 Let Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)) be a trickle graph. Notice that e
Γ = (Γ,≤, µ, (ϕ−1
x)x∈V(Γ))
is also a trickle graph which we call the dual trickle graph of Γ. On the other hand, we can define the
dual trickle group e
Tr(Γ) by the following presentation.
e
Tr(Γ)=⟨V(Γ)|xµ(x)=1 for x∈V(Γ) such that µ(x)=∞,xϕx(y)=yϕy(x) for {x,y} ∈ E(Γ)⟩.
Trickle groups and dual trickle groups are related by the following.
Proposition 2.2 Let Γ = (Γ,≤, µ, (ϕx)x∈V(Γ))be a trickle graph. Then e
Tr(e
Γ)=Tr(Γ).
Proof The group e
Tr(e
Γ) has the following presentation.
e
Tr(e
Γ)=⟨V(Γ)|xµ(x)=1 for x∈V(Γ) such that µ(x)=∞,xϕ−1
x(y)=yϕ−1
y(x)
for {x,y} ∈ E(Γ)⟩.
Let f:V(Γ)→e
Tr(e
Γ) be the map defined by f(x)=xfor all x∈V(Γ) . Let x∈V(Γ) be such that
µ(x)=∞. Then f(x)µ(x)=xµ(x)=1. Let e={x,y} ∈ E(Γ) . If x||y, then with ewe associate
the relation xy =yx in the presentation of Tr(Γ) as well as in that of e
Tr(e
Γ). So, f(x)f(y)=f(y)f(x) .
Suppose x<y. The case y<xis treated in the same way. With ewe associate the relation
yx =ϕy(x)yin the presentation of Tr(Γ). Let x′=ϕy(x). By definition of a trickle graph we have
e′={x′,y} ∈ E(Γ) and x′<y. With e′we associate the relation yϕ−1
y(x′)=x′yin the presentation
of e
Tr(e
Γ). But, since x′=ϕy(x) , this relation is also read yx =ϕy(x)y. Thus, f(y)f(x)=f(ϕy(x)) f(y).
This shows that finduces a homomorphism f: Tr(Γ)→e
Tr(e
Γ). We show in the same way that we
have a homomorphism f′:e
Tr(e
Γ)→Tr(Γ) which sends xto xfor all x∈V(Γ). It is clear that f′is the
inverse of f, hence fis an isomorphism.
Remark The proof of Proposition 2.2 proves more than what is stated: it actually proves that the
presentation of e
Tr(e
Γ) is equal to that of Tr(Γ). Nevertheless, even if the two presentations coincide, it
will be useful subsequently to call the presentation
⟨V(Γ)|xµ(x)=1 for x∈V(Γ) such that µ(x)=∞,xϕ−1
x(y)=yϕ−1
y(x) for {x,y} ∈ E(Γ)⟩
the dual presentation of Tr(Γ).
10 P Bellingeri, E Godelle and L Paris
2.2 The trickle algorithm – Results of Section 4
Let Abe a set, which we call an alphabet, and let A∗be the free monoid on A. The elements of A∗
are called words and they are written as finite sequences. The empty word is denoted by and the
concatenation of two words w1,w2∈A∗is denoted by w1·w2. A rewriting system on A∗is defined to
be a subset R⊆A∗×A∗. Let w,w′∈A∗. We set wR
→w′or simply w→w′if there exist w1,w2∈A∗
and (u,v)∈Rsuch that w=w1·u·w2and w′=w1·v·w2. More generally, we set wR∗
→w′or simply
w→∗w′if either w=w′or there exists a finite sequence w=w0,w1,...,wp=w′in A∗such that
wi−1→wifor all i∈ {1,...,p}. A word w∈A∗is said to be R -reducible if there exists w′∈A∗
such that w→w′. Otherwise we say that wis R -irreducible. The pair (A,R) is a rewriting system for
a monoid M if ⟨A|u=vfor (u,v)∈R⟩+is a monoid presentation for M. A rewriting system for
a group G is a rewriting system for Gviewed as a monoid. In particular, in this case Agenerates G
as a monoid. If (A,R) is a rewriting system for a monoid Mand w=(α1, α2, . . . , αℓ)∈A∗, then we
denote by w=α1α2. . . αℓthe element of Mrepresented by w.
Let Rbe a rewriting system on A∗. We say that Ris terminating if there is no infinite sequence {wk}∞
k=0
in A∗such that wk−1→wkfor all k∈N≥1. We say that Ris confluent if, for all u,v1,v2∈A∗such
that u→∗v1and u→∗v2, there exists w∈A∗such that v1→∗wand v2→∗w. The importance of
terminating and confluent rewriting systems comes from the following.
Theorem 2.3 (Newman [New42]) Let (A,R)be a terminating and confluent rewriting system for a
monoid M.
(1) For all w′∈A∗there exists a unique R-irreducible word w∈A∗such that w′→∗w.
(2) For all g∈Mthere exists a unique R-irreducible word w∈A∗such that g=w.
Now, we fix a trickle graph Γ = (Γ,≤, µ, (ϕx)x∈V(Γ)), and we turn to describe a rewriting system for
Tr(Γ).
The alphabet of our rewriting system is not V(Γ)⊔V(Γ)−1, as one may expect, but a more complicated
set Ω=Ω(Γ), which is generally infinite, and which is described as follows.
Throughout the paper we use the following notations. For µ∈N≥2∪ {∞} we set Zµ=Z/µZif
µ=∞and Zµ=Zif µ=∞.
The set of syllables of Γis the abstract set
S(Γ)={xa|x∈V(Γ) and a∈Zµ(x)\ {0}} .
Astratum of Γis a finite subset U={xa1
1,xa2
2,...,xap
p} ⊆ S(Γ) such that xi=xjand {xi,xj} ∈ E(Γ)
for all i,j∈ {1,...,p}such that i=j. The support of Uis supp(U)={x1,x2,...,xp} ⊆ V(Γ) and
its length is the integer p, which is denoted by lgst(U) . The empty set ∅is assumed to be a stratum
whose support is ∅. The set of strata is denoted by Ω=Ω(Γ).
The set Ωis the alphabet of our rewriting system. The elements of Ω∗are called pilings. If u=
(U1,...,Up) is a piling, then pis the length of u, which is denoted by lgpil (u).
Now, we define three operations on the strata which will be used to define our rewriting system.
Trickle groups 11
The first operation consists in removing an element from a stratum. If U={xa1
1,xa2
2,...,xap
p}is a
non-empty stratum and xai
i∈U, then we set
L(U,xai
i)=U\ {xai
i}={xa1
1,...,xai−1
i−1,xai+1
i+1,...,xap
p}.
Note that L(U,xai
i) is a stratum.
The second operation consists in “extracting” a syllable from a stratum. Let U={xa1
1,xa2
2,...,xap
p}be
a non-empty stratum and let xai
i∈U. We number the elements of Usuch that, if xj>xk, then j<k.
Then we set
γ(U,xai
i)=(ϕa1
x1◦ϕa2
x2◦ · · · ◦ ϕai−1
xi−1)(xi)ai.
It is easily seen that γ(U,xai
i) is well-defined and belongs to S(Γ) . Moreover, it will be proved in Section
4(see Lemma 4.2) that the definition of γ(U,xai
i) does not depend on the choice of the numbering of
the elements of U.
The third operation consists in “adding” a syllable to a stratum. Let U={xa1
1,xa2
2,...,xap
p} ∈ Ωand
let yb∈S(Γ). We say that ybcan be added to Uif either y∈supp(U) or {y,xi} ∈ E(Γ) for all
i∈ {1,...,p}. Suppose ybcan be added to U. If y∈ supp(U), then we set
R(U,yb)={ϕ−b
y(x1)a1, . . . , ϕ−b
y(xp)ap,yb}.
If y=xi∈supp(U) and b+ai=0 (in Zµ(y)), then we set
R(U,yb)={ϕ−b
y(x1)a1, . . . , ϕ−b
y(xi−1)ai−1, ϕ−b
y(xi+1)ai+1, . . . , ϕ−b
y(xp)ap}.
If y=xi∈supp(U) and b+ai=0 (in Zµ(y)), then we set
R(U,yb)={ϕ−b
y(x1)a1, . . . , ϕ−b
y(xi−1)ai−1,yai+b, ϕ−b
y(xi+1)ai+1, . . . , ϕ−b
y(xp)ap}.
Note that, in the third case, since y=xi=ϕ−b
y(xi), yai+bcan be replaced by ϕ−b
y(xi)ai+b. Note also
that R(U,yb) is always a stratum.
Definition Let (U,V) be a pair of strata with V=∅and let xa∈V. We set V′=L(V,xa) and
ya=γ(V,xa). We assume that yacan be added to Uand we set U′=R(U,ya). Then we say that
r=((U,V),(U′,V′)) ∈Ω∗×Ω∗is a T-transformation. In this case we write T(U,V,xa)=(U′,V′).
We denote by R1the set of T-transformations. On the other hand, we set R0={((∅), )} ⊂ Ω∗×Ω∗,
where (∅) is the piling of length 1 whose only entry is ∅and is the empty piling of length 0. Finally,
we set R=R(Γ)=R0∪ R1.
The following will be proved in Section 4.
Theorem 2.4 Let Γ = (Γ,≤, µ, (ϕx)x∈V(Γ))be a trickle graph, let Ω=Ω(Γ)be the set of strata of Γ,
and let R=R(