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Geodesic rewriting systems and pregroups∗
Volker Diekert1 Andrew J. Duncan2
Alexei Miasnikov3
1Universit¤at Stuttgart, Universit¤atsstr. 38
D-70569 Stuttgart, Germany
2Newcastle University, Newcastle upon Tyne
NE1 7RU, United Kingdom
3McGill University, Montreal, Canada, H3A 2K6
June 15, 2009
Abstract
In this paper we study rewriting systems for groups and monoids, focus-
ing on situations where nite convergent systems may be dif cult to nd or
do not exist. We consider systems which have no length increasing rules
and are con uent and then systems in which the length reducin g rules lead
to geodesics. Combining these properties we arrive at our main object of
study which we call geodesically perfect rewriting systems. We show that
these are well-behaved and convenient to use, and give several examples of
classes of groups for which they can be constructed from natural presenta-
tions. We describe a Knuth-Bendix completion process to construct such
systems, show how they may be found with the help of Stallings’ pregroups
and conversely may be used to construct such pregroups.
Contents
1 Introduction 2
∗Part of this work was begun in 2007 when the rst and third auth or where at the CRM (Centro
Recherche Matem atica, Barcelona) at the invitation of Enric Ventura.
1

Geodesic rewriting systems 2
2 Rewriting techniques 6
2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Rewriting in monoids . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Convergent rewriting systems . . . . . . . . . . . . . . . . . . . . 8
2.4 Computing with in nite systems . . . . . . . . . . . . . . . . . . 11
3 Length-reducing and Dehn systems 12
3.1 Finite length-reducing systems . . . . . . . . . . . . . . . . . . . 12
3.2 In nite length-reducing systems . . . . . . . . . . . . . . . . . . 1 3
3.3 Weight-reducing systems . . . . . . . . . . . . . . . . . . . . . . 14
4 Preperfect systems 15
4.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Geodesically perfect rewriting systems 17
5.1 Geodesic systems . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 Geodesically perfect systems . . . . . . . . . . . . . . . . . . . . 23
6 Knuth-Bendix completion for geodesically perfect systems 25
7 Examples of preperfect systems in groups 28
7.1 Graph groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.2 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.3 HNN-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.4 Free products with amalgamation . . . . . . . . . . . . . . . . . . 31
8 Stallings’ pregroups and their universal groups 32
8.1 Rewriting systems for universal groups . . . . . . . . . . . . . . . 34
8.2 Characterisation of pregroups in terms of geodesic systems . . . . 38
1 Introduction
A presentation of a group or monoid may be thought of as a rewriting system
which, in certain cases may give rise to algorithms for solving classical algorith-
mic problems. For example if the rewriting system is nite an d convergent (that
is con uent and terminating) then it can be used to solve the w ord problem and
to nd normal forms for elements of the group. This is one reas on for the impor-
tance of convergent rewriting systems in group theory. However there are many
groups for which the natural presentations do not give rise to convergent rewrit-
ing systems, but which are none the less well behaved, algorithmically tractable
groups. In this paper we investigate properties of rewriting systems, which are not