Gromov boundaries as Markov compacta
Abstract
We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of ∂ G, which is quasi-G-invariant wrt. the ball N-type (defined by Cannon) for N sufficiently large. We also ensure certain additional properties for the inverse system representing ∂ G, leading to a finite description which defines it uniquely. By defining a natural metric on the inverse limit Kn and proving it to be bi-Lipschitz equivalent to an accordingly chosen visual metric on ∂ G, we prove that our construction enables providing a simplicial description of the natural quasi-conformal structure on ∂ G. We also point out that the initial system of covers can be modified so that all the simplexes in the resulting inverse system are of dimension less than or equal to ∂ G. We also generalize --- from the torsion-free case to all finitely generated hyperbolic groups --- a theorem guaranteeing the existence of a finite representation of ∂ G of another kind, namely a semi-Markovian structure (which can be understood as an analogue of the well-known automatic structure of G itself).