Inventiones mathematicae

, Volume 185, Issue 2, pp 395–419 | Cite as

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

  • Kai-Uwe Bux
  • Kevin Wortman


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.


Morse Function Reduction Theory Irreducible Factor Index Subgroup Finiteness Property 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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