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Kapovich, Michael and Leeb, Bernhard (2018): Finsler bordifications of symmetric and certain locally symmetric spaces. In: Geometry & Topology, Vol. 22, No. 5: pp. 2533-2646

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Abstract

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-withcorners, of locally symmetric spaces X/Gamma for arbitrary discrete subgroups Gamma < G . These bordifications result from attaching Gamma-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 Haissinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.

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