Automata and growth functions of Coxeter groups

• Main Author: Avasjö, Agneta
• Format: Others
• Language: English
• Published: KTH, Matematik 2004
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-1814; http://nbn-resolving.de/urn:isbn:91-7283-852-3

The aim of this thesis is to study, in some detail,properties of growth functions and geodesic growth functionsfor Coxeter groups. To do this, we use the fact that allCoxeter groups, which can be defined by some simple rules on apresentation by generators and relators, are described byformal languages which satisfy rather strong finitaryconditions. By connecting the context of groups with that offormal languages and constructing finite state automata for thelanguages N(G, S) and L(G, S) we make explicit algorithmiccomputations of the corresponding growth functions of the groupG. As a test-case we choose the subclass of triangle groups,which are defined in a purely geometric way as groups generatedby reflections with respect to the sides of a triangle. Thetheorems and the methods shown are however valid for allCoxeter groups. The construction of the automatons is based ona representation of a Coxeter group by linear transformationsacting on a vector space. The key notion here is that of a rootsystem. We demonstrate that the growth series and the growthseries of geodesics associated with a Coxeter system can bothbe given by rational expressions. Triangle groups (except for a finite number) are naturallyorganized into a few infinite series, and we were able toperform our computations for these infinite series, with one orseveral parameter tending to infinity. We give graphicalrepresentations of the constructed automata as well as resultsof numerical computations of the corresponding growthfunctions.