Volume 22, issue 1 (2018)

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Detecting sphere boundaries of hyperbolic groups

Benjamin Beeker and Nir Lazarovich

Geometry & Topology 22 (2018) 439–470
Abstract

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-$1$ surface subgroups has $\partial G\cong {\mathbb{S}}^{2}\phantom{\rule{0.3em}{0ex}}$. By work of Markovic (2013), our result gives a new characterization of virtually fundamental groups of hyperbolic $3$–manifolds.

Keywords
$\mathrm{CAT}(0)$ cube complexes, hyperbolic groups, hyperbolic $3$–manifolds
Mathematical Subject Classification 2010
Primary: 20F65, 20F67, 20H10