NDLI: Floyd maps for relatively hyperbolic groups

Content Provider	Springer Nature Link
Author	Gerasimov, Victor
Copyright Year	2012
Abstract	Let $${\mathbf{delta}_{\mathcal S,\lambda}}$$ denote the Floyd metric on a discrete group G generated by a finite set $${\mathcal S}$$ with respect to the scaling function f n = λ n for a positive λ < 1. We prove that if G is relatively hyperbolic with respect to a collection $${\mathcal P}$$ of subgroups then there exists λ such that the identity map $${G\to G}$$ extends to a continuous equivariant map from the completion with respect to $${\mathbf{\delta}_{\mathcal S,\lambda}}$$ to the Bowditch completion of G with respect to $${\mathcal P}$$ . In order to optimize the proof and the usage of the map theorem we propose two new definitions of relative hyperbolicity equivalent to the other known definitions. In our approach some “visibility” conditions in graphs are essential. We introduce a class of “visibility actions” that contains the class of relatively hyperbolic actions. The convergence property still holds for the visibility actions. Let a locally compact group G act on a compactum Λ with convergence property and on a locally compact Hausdorff space Ω properly and cocomactly. Then the topologies on Λ and Ω extend uniquely to a topology on the direct union $${T=\Lambda{\sqcup}\Omega}$$ making T a compact Hausdorff space such that the action $${G{\curvearrowright}T}$$ has convergence property. We call T the attractor sum of Λ and Ω.
Ending Page	1399
Page Count	39
Starting Page	1361
File Format	PDF
ISSN	1016443X
e-ISSN	14208970
Journal	Geometric and Functional Analysis
Issue Number	5
Volume Number	22
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-09-22
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Topological methods in group theory Convergence group action Hyperbolic groups and nonpositively curved groups Analysis Kleinian groups Relatively hyperbolic group Geometric group theory Floyd completion Locally compact group General properties and structure of locally compact groups
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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