Compactifications of S-arithmetic Quotients for the Projective General Linear Group

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 188)


Let F be a global field, let S be a nonempty finite set of places of F which contains the archimedean places of F, let \(d\geqslant 1\), and let \(X =\prod _{v\in S} X_v\) where \(X_v\) is the symmetric space (resp., Bruhat-Tits building) associated to \({{\mathrm{PGL}}}_d(F_v)\) if v is archimedean (resp., non-archimedean). In this paper, we construct compactifications \(\Gamma \backslash \bar{X}\) of the quotient spaces \(\Gamma \backslash X\) for S-arithmetic subgroups \(\Gamma \) of \({{\mathrm{PGL}}}_d(F)\). The constructions make delicate use of the maximal Satake compactification of \(X_v\) (resp., the polyhedral compactification of \(X_v\) of Gérardin and Landvogt) for v archimedean (resp., non-archimedean). We also consider a variant of \(\bar{X}\) in which we use the standard Satake compactification of \(X_v\) (resp., the compactification of \(X_v\) due to Werner).


Primary 14M25 Secondary 14F20 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUCLALos AngelesUSA

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