The limit sets of Coxeter groups
良介 嶺山 · OUKA (Osaka University Knowledge Archive) (Osaka University) · 2014
Future prospectsThe author studies the action of groups on metric spaces from the asymptotical view point.In this thesis, we deal with actions of Coxeter groups on the hyperbolic space to obtain information on the roots of the given Coxeter group via the limit set.In particular the author would like to derive a kind of rigidity property of Coxeter groups.We already observed a rigidity at infinity in a different situation.Indeed in a joint work with Miyachi [27] the author has shown Royden's theorem which states that any biholomorphic automorphism on Teichmüller space is induced by an orientation preserving homeomorphism on the base surface.Our proof is accomplished by studying the behavior of biholomorphic automorphisms at infinity of Teichmüller space to get the rigidity.For the future, we can consider the following:The relation between the Hausdorff dimension and the growth rate of the given Coxetr group.Actually, the growth function is nothing but the Poincaré series.From this point of view, for a Coxeter group of rank 3 we can see how the deformation of the action affects the Hausdorff dimension of the limit set ([28]).groups together with Yuriko Umemoto which are origin of our collaboration described in Chapter 2. I would also like to express my gratitude to all my colleagues, especially my room mates, Yuta Wakasugi, Ryoichi Kase, Mai Fujita and Yuuki Shiraishi for teaching me their exciting mathematics.Their encouragement helped me so many times not only for the mathematics.The list of persons who I want to give my thank's is never ending.For the last, I heartily thank my family for their unremitting encouragement and moral supports without words.CHAPTER 1 Normalized reflections and Coxeter groupsIn this chapter we collect some notations and definitions of notions used throughout this thesis.Especially a crucial assumption (Assumption 1.1) is introduced.