an:06125049
Zbl 1276.20050
Gerasimov, Victor
Floyd maps for relatively hyperbolic groups
EN
Geom. Funct. Anal. 22, No. 5, 1361-1399 (2012).
00310217
2012
j
20F65 20F67 22D05 57M07 20F05 30F40
Floyd completions; relatively hyperbolic groups; convergence group actions; locally compact groups; Cayley graphs; geometrically finite Kleinian groups
Given a finitely generated group \(G\) and a finite generating set, the edges in the Cayley graph have length \(1\). W. J. Floyd conformally deformed the metric to get a metric space with finite diameter. Let \(f\colon\mathbb N\to (0,\infty)\) be a non-increasing function with \(\sum_{n=0}^\infty f(n)<\infty\) such that, for any \(k\geq 1\), there are \(M,N>0\) so that \(Mf(r)\leq f(kr)\leq Nf(r)\) for all \(r\). Define a new length metric on the Cayley graph by giving an edge \(e\) length \(f(n)\) if the distance from \(e\) to the identity element is \(n\). This new metric space \(\Gamma_f\) has finite diameter. Its completion \(\overline\Gamma_f\) is a Floyd completion and \(\partial_f G=\overline\Gamma_f\setminus\Gamma_f\) is a Floyd boundary. The left translation action of \(G\) on itself induces actions on \(\overline\Gamma_f\) and \(\partial_fG\) by biLipschitz maps.
Floyd proved that if \(G\) is a geometrically finite Kleinian group and \(f(n)=\frac{1}{n^2+1}\), then there is a continuous equivariant map from the Floyd boundary to the limit set. The main result of the current paper generalizes this result to the case of relatively hyperbolic groups. Given a relatively hyperbolic group \(G\), there is some \(0<\lambda<1\), such that for the function \(f(n)=\lambda^n\), there is a continuous equivariant map from the Floyd boundary \(\partial_fG\) to the Bowditch boundary. This in particular implies the Floyd boundary is non-trivial.
To facilitate the proof of the main theorem, the author proposes two new definitions of relatively hyperbolic groups, which are equivalent to the known ones. They serve as bridges between the geometric definition (actions on fine graphs) and the dynamics definition (convergence action).
Another interesting result is the so called Attractor Sum Theorem. It says the following: given a convergence action of a locally compact group \(G\) on a compactum \(\Lambda\), and a proper cocompact action of \(G\) on a locally compact Hausdorff space \(\Omega\), there is a unique topology on the disjoint union \(T:=\Lambda\cup\Omega\) extending the topologies on \(\Lambda\) and \(\Omega\), such that \(T\) is compact Hausdorff and the action of \(G\) on \(T\) is also a convergence action.
Xiangdong Xie (Bowling Green)