# Hyperbolic structures on groups

@article{Abbott2019HyperbolicSO,
title={Hyperbolic structures on groups},
author={Carolyn R. Abbott and Sahana Balasubramanya and Denis V. Osin},
journal={Algebraic \& Geometric Topology},
year={2019}
}
• Published 14 October 2017
• Mathematics
• Algebraic & Geometric Topology
For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $\mathcal{H}(G)$, 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…

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