# A SIMPLE GRAMMAR FOR GENERATING COCOMPACT FUCHSIAN GROUPS

**ABSTRACT** In this paper, we present a regular grammar that generates unique representatives of all elements in a cocompact Fuchsian group (CFG) from a given trivalent presentation (to be defined below). This grammar is the simplest possible in the sense that it possesses the fewest productions.

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- Quarterly Journal of Mathematics - QUART J MATH. 01/1982; 33(4):451-461.
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##### Article: Théorie des groupes fuchsiens

Acta Mathematica 01/1970; 1(1):1-62. · 2.71 Impact Factor

Page 1

International Journal of Pure and Applied Mathematics

————————————————————————–

Volume 46No. 22008, 165-170

Invited Lecture Delivered at

Forth International Conference of Applied Mathematics

and Computing (Plovdiv, Bulgaria, August 12–18, 2007)

A SIMPLE GRAMMAR FOR GENERATING

COCOMPACT FUCHSIAN GROUPS

O. Michael Melko1 §, Luc Patry2

1Department of Mathematics

Northern State University

1200 South Jay Str., Aberdeen, South Dakota 57401-7199, USA

e-mail: mike.melko@northern.edu

2Department of Mathematics

University of Arkansas

Pine Bluff, AR 71601, USA

e-mail: patryl@uapb.edu

Abstract: In this paper, we present a regular grammar that generates unique

representatives of all elements in a cocompact Fuchsian group (CFG) from a

given trivalent presentation (to be defined below). This grammar is the simplest

possible in the sense that it possesses the fewest productions.

AMS Subject Classification: 53A35, 30F35, 20F05, 68Q42

Key Words: hyperbolic geometry, Fuchsian groups, formal languages

1. Introduction

A Fuchsian group G is a discrete subgroup of the isometry group of the hyper-

bolic plane H and is said to be cocompact if the corresponding quotient space

M = H/G is a compact hyperbolic surface (i.e., a surface of constant negative

curvature). Poincar´ e [4] was the first to systematically study these groups and

their geometry. His method is to associate a finite presentation of G with a

fundamental domain of the projection π : H → M. In this way, G is seen to

Received:August 17, 2007

c ? 2008, Academic Publications Ltd.

§Correspondence author

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166O.M. Melko, L. Patry

be isomorphic to the fundamental group of the quotient. Although it is rela-

tively straightforward to produce a presentation of a Fuchsian group given a

Poincar´ e domain, it is a nontrivial matter to uniquely represent each element

of the group as a product of the generators in a computationally efficient man-

ner. This problem falls under the purview of the modern theory of automatic

groups, for which Epstein, et al [1] is the standard reference.

In this paper, we summarize some results that will be presented in greater

detail in [3]. We will begin by reviewing some basic facts about the topology

and geometry of Poincar´ e domains associated with a compact hyperbolic sur-

face. We will then present a regular grammar for generating the elements of a

Fuchsian group given a particular kind of presentation, and we will state a a

recursion formula for counting the number of words of a given length.

We define an abstract graph to be an ordered pair A = (A0,A1), where

A0is a finite set whose elements we may label with positive integers, and the

elements of A1 are undirected incidence relations between pairs of elements

in A0.Formally, A1 ⊆ A0× A0/ ∼, where (x′,y′) ∼ (x,y) if and only if

(x′,y′) = (x,y) or (x′,y′) = (y,x).We further define an embedding of an

abstract graph A into a hyperbolic surface M to be a map ν : A → M. We

write vifor the point v(i) in M and require that each edge {i,j} be mapped

onto a geodesic segment in M whose endpoints are {vi,vj}. We further require

that distinct edges in M are either disjoint or intersect only at vertices.

Let E := ν(A) denote the image of an abstract graph A into M under

an embedding v. Thus E = (E0,E1), where E0 denotes the set of vertices

v(A0), and E1 denotes the corresponding set of edges. Then E is naturally

a one-dimensional CW-complex. Using Σk(X) to denote the k-skeleton of a

CW-complex X, we will say that E is a Poincar´ e graph if E2:= M \ Σ1(E) is

a contractible cell, and the valence of each vertex is at least 3. The extended

complexˆE := (E0,E1,E2) is a decomposition of M into a CW-complex with

a single two-dimensional cell. We further say that A is an abstract admissible

graph if it admits an embedding ν(A) into M that is a Poincar´ e graph. We

refer to such an embedding as an admissible embedding.

We say that two graphs are homeomorphic if there is a one-to-one correspon-

dence between their vertices and edges that preserves their respective incidence

relations. Two graphs are homotopically equivalent if one can be obtained from

the other through a series of edge contractions and expansions. We refer to a

graph that has only one vertex as a one-vertex graph, and we refer to a graph

in which every vertex has valence 3 (the other extreme) as a trivalent graph. It

is clear that every Poincar´ e graph is homotopically equivalent to a one-vertex

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A SIMPLE GRAMMAR FOR GENERATING...167

Poincar´ e graph through a series of edge contractions. Similarly, by means of

edge expansions at vertices with valence greater than 3, every Poincar´ e graph

can be expanded into a trivalent Poincar´ e graph. Note that the number of

cycles in a Poincar´ e graph must be invariant under homotopy and is equal to

2p, where p denotes the genus of M. By means of the Euler characteristic, it is

easily verified that, for any Poincar´ e graph E, we must have 1 ≤ |E0| ≤ 4p−2.

In particular, |E0| = 1 ⇒ |E1| = 2p and |E0| = 4p−2 ⇒ |E1| = 6p−3. For the

case p = 2, we have |E0| = 6 and |E1| = 9.

We refer to the number of vertices |E0| in an admissible graph E as the

rank of E, and we say that E has full rank if |E0| = 4p−2. It is easily verified

that a Poincar´ e graph E has full rank if and only if it is trivalent. It is shown in

[2] that there are five homeomorphically distinct trivalent abstract admissible

graphs with eight homotopically distinct admissible embeddings into M for the

case p = 2. As the authors point out, the number of homeomorphically distinct

abstract admissible graphs grows very rapidly with the genus. The space of

Poincar´ e graphs in a compact hyperbolic surface M may be geometrized by

means of the arc-length functional L. It will be shown in [3] that this space is

a stratified Riemannian space on which L is Morse.

Now let E be a Poincar´ e graph in M. Then the open cell M \ E may be

isometrically lifted to an open set P′

1⊆ H. This lifting is unique up to an

isometry of H. The closure P1 of P′

1in H is a geodesic polygon in H. We

shall refer to a polygon in H that arises in this way as a Poincar´ e domain

corresponding to E in M. The polygon P1necessarily has an even number of

edges with a pairing of sides of equal length that arises from projecting the

boundary of P1onto E. Suppose now that {e1,...,en,e∗

the edges of P1, where ekis paired with e∗

we define xk to be the unique isometry of H mapping ek onto e∗

X := {x1,...,xn} defines a set of generators for a Fuchsian group G. The

set R of relations is found by determining all inequivalent minimal products

of generators that fix a vertex of P1 and one of its edges.

corresponding presentation as Γ = ?X|R?. Note that the elements of G are in

one-to-one correspondence with tiles in a tessellation of H that are congruent to

the base tile P1. We will say that Γ is a one-vertex presentation if it arises from

a one-vertex Poincar´ e graph, and we will say that it is a trivalent presentation

if it arises from a trivalent Poincar´ e graph. One-vertex presentations, in which

there is one relation consisting of a product of commutators, are the ones most

commonly discussed in the literature. In what follows, we consider only trivalent

presentations. For genus p, there are 4p − 2 inequivalent relations of the form

a · b · c = 1 for some choice of a,b,c ∈ X ∪ X−1, one for each “orbit class” of

1,...,e∗

n} is a list of

kfor 1 ≤ k ≤ n. For each 1 ≤ k ≤ n,

k. The set

We denote the

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168O.M. Melko, L. Patry

vertices in P1.

Our objective in the next section is to describe a generative grammar that

solves Problem 1. Due to lack of space, we refer the reader to [3] for a detailed

derivation of this result.

Problem 1.

Fuchsian group G, generate a set of words over the alphabet K = X∪X−1∪{1}

that are unique representatives of all group elements in G. Here, X−1denotes

the set consisting of the multiplicative inverses of elements in X.

Given a trivalent presentation Γ = ?X|R? of a cocompact

2. A Regular Grammar for Trivalent Presentations

We begin this section by reviewing some preliminaries from the theory of formal

languages. We refer to a finite set K as an alphabet, and we refer to its elements

as symbols. It is convenient to augment K with the symbol φ, which we call

the null symbol, and we write Kφ= K ∪ {φ} for the extended alphabet. A

word x = x1···xnover K is an ordered sequence of symbols xi∈ K. A word

x = x1···xn+1 is said to be null-terminated if xi ∈ K for 1 ≤ i ≤ n and

xn+1 = φ. We sometimes express the fact that a symbol y ∈ K appears in

a word x by writing y ∈ x. The concatenation of two words x = x1···xn

and y = y1···ymwill be written as x · y := x1···xny1···ym. We refer to the

number of symbols in a word x as its length, and we denote the length of x by

|x|. This length function is additive in the sense that |x · y| = |x| + |y|. We

refer to x[i] := xi as the symbol at position i in the word x. Similarly, for

1 ≤ i < j ≤ |x|, we write x[i,j] := xi···xj, x[i,j) := xi···xj−1, x(i,j) :=

xi+1···xj−1, and x(i,j] := xi+1···xj, for the corresponding parts of x. If

1 ≤ j < i ≤ |x|, we understand x[i,j], x[i,j), etc., to be the word obtained by

wrapping around the ends of the word x. Thus, if x = abcdef, for example,

we have x[5,2] := efab. We define the increment operator + on a word x of

length n by the rule xi+ = xi+1for 1 ≤ i < n, and we set xn+ = x1. In the

same fashion, we define the decrement operator − by the rule xi− = xi−1for

1 < i ≤ n and x1− = xn. Note that, if a word x contains no repeated symbols,

we may unambiguously identify a symbol with its position in that word. Thus,

in the previous example, x[e,b] = x[5,2], e+ = f, e++ = a, etc.

The set of all words over K, together with the null word φK, is a semigroup

under concatenation and will be denoted by K∗. A subset L ⊆ K∗of K∗is a

(formal) language over K. We may concatenate every word in a language on

the right with the null symbol. The language derived in this way consists of

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A SIMPLE GRAMMAR FOR GENERATING...169

null-terminated words and will be denoted by L′(K) = L(K) · φ. This artifice

is useful for normalizing the language acceptor automaton defined in [3] and

associated with the grammar displayed in Table 1. It also corresponds naturally

to certain data structures, such as null-terminated strings in the C programming

language.

Useful languages are often described in terms of generative grammars, which

provide rules for generating words. There is a hierarchy of such grammars, but

we only require the most restrictive kind, which we now define.

Definition 2. A (right) regular generative grammar is a pair G = (Σ,Π)

consisting of a set of symbols Σ and a collection of rewriting rules or productions

Π that satisfy the following conditions:

(i) There are two special subsets ΣNand ΣT of Σ such that Σ = ΣN∪ ΣT

and ΣN∩ ΣT= ∅. We call ΣNthe set of nonterminal symbols and ΣT the set

of terminal symbols.

(ii) There is a distinguished symbol σ0∈ ΣN called the initial symbol or

start symbol. This symbol is used to begin the derivation of words in a language.

(iii) The productions in Π are of the form X → wY or X → w, where

X,Y ∈ ΣNand w ∈ Σ∗

T.

We will write L(Σ,Π) or L(G) for the language over ΣT that is generated

by the grammar G = (Σ,Π).

We now associate a cyclic Wick’s form W with a presentation Γ of a

Fuchsian group. First, choose a counterclockwise ordering of the edges of a

Poincar´ e domain P1and label them accordingly as {e1,e2,...,e2n}. Now set

W = w1w2w3···w2n, where wk∈ X ∪X−1is the isometry of H that sends the

edge ekto the other edge in its pairing.

Now imagine that we are building a tessellation of H by repeated application

of the elements of X ∪X−1to P1in such a way that none of the resulting tiles

overlap. We say that a tile is at level l if it takes a minimum of l operations

to arrive at its destination, and we further say that it is of type k if it has a

common edge with exactly k tiles at one lower level. Because the presentation

is trivalent, tiles can only be of type 0, 1, or 2. The first tile P1is the only tile of

type 0, and all tiles at level 1 are of type 1. We use the symbol Py, resp. Py,y+,

to denote a type 1, resp. type 2, tile that is arrived at from a previous tile by

means of an application of y ∈ W. Our nonterminal symbols are now taken

to be ΣN:= {Py|y ∈ W · 1} ∪ {Py,y+|y ∈ W}, and our terminal symbols are

taken to be ΣT:= Kφ= X∪X−1∪{1,φ}. The corresponding productions that

generate the group language are given in Table 1. Proposition 3 gives us a way

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170O.M. Melko, L. Patry

to count tiles at a given level (or words of a given length). For a closed-form

formula, see [3].

Proposition 3. Let P1be an m-sided Poincar´ e domain, and let pk,ldenote

the number of tiles of type k at level l. Then p1,1= m and p2,1= 0, and, for

l ≥ 1,

p1,l+1= (m − 5)p1,l+ (m − 6)p2,l,p2,l+1= p1,l+ p2,l.

(i)σ0→ yPy

y ∈ W · 1

(ii)P1→ φ

(iii)Py→ zPz

Py→ zPz,z+

Py→ φ

z ∈ (y++,y−−)

z = y++

y ∈ W

(iv)Py,y+→ zPz

Pz,z+→ zPz,z+

Py,y+→ φ

z ∈ (y+++,y−−)

z = y+++

y ∈ W

Table 1: Productions for generating the elements of a cocompact Fuch-

sian group

References

[1] David B.A. Epstein, et al Word Processing in Groups, Jones and Bartlett

Publishers (1992).

[2] Troels Jørgenson, Marjatta N¨ a¨ at¨ anen, Surfaces of genus 2: Generic funda-

mental domains, Quart.J. Math. Oxford, 33 (1982), 451-461.

[3] O. Michael Melko, Luc Patry, The hyperbolic Steiner problem, Teichm¨ uller

space, and the efficient computation of cocompact Fuchsian groups, In

Preparation.

[4] H. Poincar´ e, Th´ eorie des groupes fuchsiens, Acta Mathematica, 1 (1882),

1-62.

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