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Constructing thin subgroups of $\mathrm{SL}(n+1,\mathbb{R})$ via bending

Samuel A Ballas and Darren D Long

Algebraic & Geometric Topology 20 (2020) 2071–2093
Abstract

We use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite-volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it is possible to find infinitely many noncommensurable lattices in SL(n + 1, ) that contain a thin subgroup isomorphic to a finite-index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all noncocompact arithmetic lattices as well as all cocompact arithmetic lattices when n is even.

Keywords
thin groups, bending, projective structures, arithmetic groups
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 22E40
References
Publication
Received: 22 March 2019
Revised: 28 October 2019
Accepted: 21 November 2019
Published: 20 July 2020
Authors
Samuel A Ballas
Department of Mathematics
Florida State University
Tallahassee, FL
United States
Darren D Long
Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA
United States