The structure of relatively hyperbolic groups in convex real projective geometry

Islam, Mitul; Zimmer, Andrew

doi:10.2140/agt.2025.25.5503

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ISSN (electronic): 1472-2739

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The structure of relatively hyperbolic groups in convex real projective geometry

Mitul Islam and Andrew Zimmer

Algebraic & Geometric Topology 25 (2025) 5503–5539

DOI: 10.2140/agt.2025.25.5503

Abstract

We prove a general structure theorem for relatively hyperbolic groups (with arbitrary peripheral subgroups) acting naive convex cocompactly on properly convex domains in real projective space. We also establish a characterization of such groups in terms of the existence of an invariant collection of closed unbounded convex subsets with good isolation properties. This is a real projective analogue of results of Hindawi–Hruska–Kleiner for $C A T (0)$ spaces. We also obtain an equivariant homeomorphism between the Bowditch boundary of the group and a quotient of the ideal boundary.

Keywords

relatively hyperbolic groups, convex cocompact groups, discrete subgroups of Lie groups

Mathematical Subject Classification

Primary: 20F65, 20F67, 22E40, 57N16, 57S20

References

Publication

Received: 13 July 2023

Revised: 23 August 2024

Accepted: 14 January 2025

Published: 18 December 2025

Authors

Mitul Islam
Mathematisches Institut Heidelberg Germany	Max Planck Institute for Mathematics in the Sciences Leipzig Germany

Andrew Zimmer
Department of Mathematics University of Wisconsin–Madison Madison, WI United States

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Volume 25, issue 9 (2025)