Arithmetic subgroups of Chevalley group schemes over function fields I: quotients of the Bruhat-Tits building by \P\-arithmetic subgroups

2022/07/13 by Bravo, Claudio, Loisel, Benoit
#11R58 #20E42 (primary) 14H05 #20G30 #20H25 (secondary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2207.06546

Abstract

Let G be a reductive Chevalley group scheme (defined over ℤ). Let C be a smooth, projective, geometrically integral curve over a field \mathbbF. Let P be a closed point on C. Let A be the ring of functions that are regular outside \lbrace P \rbrace. The fraction field k of A has a discrete valuation ν=ν_P: k^× → ℤ associated to P. In this work, we study the action of the group G(A) of A-points of G on the Bruhat-Tits building X=X(G,k,ν_P) in order to describe the structure of the orbit space G(A)\backslash X. We obtain that this orbit space is the ``gluing'' of a closed connected CW-complex with some sector chambers. The latter are parametrized by a set depending on the Picard group of C \smallsetminus \P\ and on the rank of G. Moreover, we observe that any rational sector face whose tip is a special vertex contains a subsector face that embeds into this orbit space.

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