Topological boundaries of connected graphs and Coxeter groups

Abstract

We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov’s hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter systems and are intimately related to the corresponding Iwahori–Hecke algebras. We study this connection by considering dynamical properties of the induced action of the Coxeter group.