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For every group G, we define the set of hyperbolic structures on G, denoted by ℋ(G), which consists of equivalence classes of (possibly infinite) generating sets of G such that the corresponding Cayley graph is hyperbolic; two generating sets of G are equivalent if the corresponding word metrics on G are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded G–actions on hyperbolic spaces. We are especially interested in the subset Aℋ(G) ⊆ℋ(G) of acylindrically hyperbolic structures on G, ie hyperbolic structures corresponding to acylindrical actions. Elements of ℋ(G) can be ordered in a natural way according to the amount of information they provide about the group G. The main goal of this paper is to initiate the study of the posets ℋ(G) and Aℋ(G) for various groups G. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of G, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.
Keywords: hyperbolic structures, group actions on hyperbolic spaces, acylindrical actions
Abbott, Carolyn 1 ; Balasubramanya, Sahana 2 ; Osin, Denis 3
@article{10_2140_agt_2019_19_1747,
author = {Abbott, Carolyn and Balasubramanya, Sahana and Osin, Denis},
title = {Hyperbolic structures on groups},
journal = {Algebraic and Geometric Topology},
pages = {1747--1835},
publisher = {mathdoc},
volume = {19},
number = {4},
year = {2019},
doi = {10.2140/agt.2019.19.1747},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1747/}
}
TY - JOUR AU - Abbott, Carolyn AU - Balasubramanya, Sahana AU - Osin, Denis TI - Hyperbolic structures on groups JO - Algebraic and Geometric Topology PY - 2019 SP - 1747 EP - 1835 VL - 19 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1747/ DO - 10.2140/agt.2019.19.1747 ID - 10_2140_agt_2019_19_1747 ER -
%0 Journal Article %A Abbott, Carolyn %A Balasubramanya, Sahana %A Osin, Denis %T Hyperbolic structures on groups %J Algebraic and Geometric Topology %D 2019 %P 1747-1835 %V 19 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1747/ %R 10.2140/agt.2019.19.1747 %F 10_2140_agt_2019_19_1747
Abbott, Carolyn; Balasubramanya, Sahana; Osin, Denis. Hyperbolic structures on groups. Algebraic and Geometric Topology, Tome 19 (2019) no. 4, pp. 1747-1835. doi: 10.2140/agt.2019.19.1747
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