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Finsler bordifications of symmetric and certain locally symmetric spaces

Michael Kapovich and Bernhard Leeb

Geometry & Topology 22 (2018) 2533–2646

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X = GK 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 XΓ 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.

discrete groups, Finsler geometry
Mathematical Subject Classification 2010
Primary: 20F65, 22E40, 53C35
Secondary: 51E24, 53B40
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
Michael Kapovich
MK: Department of Mathematics
University of California, Davis
Davis, CA
United States
MK: Korea Institute for Advanced Study
South Korea
Bernhard Leeb
BL: Mathematisches Institut
Universität München