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Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups

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Abstract

Let \(\mathbf{G}(\mathcal{O}_{S})\) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of \(\mathbf{G}(\mathcal{O}_{S})\) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of \(\mathbf{G}(\mathcal{O}_{S})\) established in an earlier paper is sharp in this case.

The geometric analysis underlying our result determines the connectivity properties of horospheres in thick Euclidean buildings.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, 33501, Bielefeld, Germany

    Kai-Uwe Bux

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112-0090, USA

    Kevin Wortman

Authors
  1. Kai-Uwe Bux
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  2. Kevin Wortman
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Correspondence to Kai-Uwe Bux.

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The second author was partially supported by NSF grant DMS-0750032.

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Bux, KU., Wortman, K. Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups. Invent. math. 185, 395–419 (2011). https://doi.org/10.1007/s00222-011-0311-1

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  • DOI: https://doi.org/10.1007/s00222-011-0311-1

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