Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X = G ∕ K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$ –invariant Finsler metric on $X$ . As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X ∕ Γ$ for arbitrary discrete subgroups $Γ < G$ . These bordifications result from attaching $Γ$ –quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion-free case, to a question of Haïssinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.

Keywords

discrete groups, Finsler geometry

Mathematical Subject Classification 2010

Primary: 20F65, 22E40, 53C35

Secondary: 51E24, 53B40

References

Publication

Received: 14 March 2016

Revised: 6 July 2017

Accepted: 3 February 2018

Published: 1 June 2018

Proposed: Bruce Kleiner

Seconded: Jean-Pierre Otal, Tobias H Colding

Authors

Michael Kapovich
MK: Department of Mathematics University of California, Davis Davis, CA United States	MK: Korea Institute for Advanced Study Seoul South Korea

Bernhard Leeb
BL: Mathematisches Institut Universität München München Germany

Volume 22, issue 5 (2018)

Michael Kapovich and Bernhard Leeb

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors