Abstract
For every group , we define the set of hyperbolic structures on , denoted by , which consists of equivalence classes of (possibly infinite) generating sets of such that the corresponding Cayley graph is hyperbolic; two generating sets of are equivalent if the corresponding word metrics on are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded –actions on hyperbolic spaces. We are especially interested in the subset of acylindrically hyperbolic structures on , ie hyperbolic structures corresponding to acylindrical actions. Elements of can be ordered in a natural way according to the amount of information they provide about the group . The main goal of this paper is to initiate the study of the posets and for various groups . 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 , and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.
Citation
Carolyn Abbott. Sahana H Balasubramanya. Denis Osin. "Hyperbolic structures on groups." Algebr. Geom. Topol. 19 (4) 1747 - 1835, 2019. https://doi.org/10.2140/agt.2019.19.1747
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