Compactifications of S-arithmetic quotients for the projective general linear group

TL;DR

This paper develops new compactifications for S-arithmetic quotients of PGL_d, combining techniques from reductive Borel-Serre spaces and polyhedral compactifications, with potential applications in arithmetic geometry.

Contribution

It introduces a novel construction of compactifications for S-arithmetic quotients of PGL_d, integrating archimedean and non-archimedean methods.

Findings

Constructed compactifications using reductive Borel-Serre spaces and polyhedral methods.

Provided a framework applicable to various potential applications in arithmetic geometry.

Extended the theory of compactifications to include S-arithmetic quotients of PGL_d.

Abstract

Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications.