Constructing thin subgroups of SL(n + 1, ℝ) via bending

Ballas, Samuel A; Long, Darren D

doi:10.2140/agt.2020.20.2071

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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

DOI: 10.2140/agt.2020.20.2071

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 $SL (n + 1, ℝ)$ 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

Volume 20, issue 4 (2020)