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The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields

Gebhard Böckle, Sriram Chinthalagiri Venkata

Abstract

Let $K$ be a function field in positive characteristic, $\infty$ be a fixed place of $K$ and $K_\infty$ be the completion of $K$ at $\infty$. By the work of Serre, it is well known that, for a suitable arithmetic subgroup $Γ\subset GL_2(K)$, the $Γ$-unstable region of the Bruhat-Tits tree for $GL_2(K_\infty)$ is naturally homotopy equivalent to the spherical Tits building for $GL_2(K)$. Grayson, following Quillen's ideas, generalizes this homotopy equivalence to the non-semistable part of the Bruhat-Tits building for $GL_r(K_\infty)$. Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the $Γ$-unstable region, for $Γ\subset GL_r(K)$ a principal congruence subgroup.

The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields

Abstract

Let be a function field in positive characteristic, be a fixed place of and be the completion of at . By the work of Serre, it is well known that, for a suitable arithmetic subgroup , the -unstable region of the Bruhat-Tits tree for is naturally homotopy equivalent to the spherical Tits building for . Grayson, following Quillen's ideas, generalizes this homotopy equivalence to the non-semistable part of the Bruhat-Tits building for . Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the -unstable region, for a principal congruence subgroup.
Paper Structure (16 sections, 32 theorems, 53 equations)

This paper contains 16 sections, 32 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Preliminaries on simplicial complexes
  4. Geometric realization, ($G$-)CW complexes and equivariant Whitehead's theorem
  5. Homotopy properties of $G$-simplicial complexes
  6. Simplicial complexes on $G$-posets and their (equivariant) homotopy properties
  7. $G$-equivariant properties
  8. Bruhat-Tits building
  9. Congruence subgroups
  10. $\mathcal{B}_\bullet(W)$ as a building of vector bundles
  11. Labeling the vertices of $\mathcal{X}_\bullet(P)$.
  12. Stabilizers of simplices
  13. An equivariant homotopy between the unstable subcomplex and the (rational) Tits building(after Grayson-Quillen)
  14. Contractible subcomplexes of the $\Gamma$-unstable complex
  15. The $\Gamma_P$-equivariant homotopy $\mathcal{B}_\bullet(W)^{\Gamma\textrm{-}\mathrm{un}} \rightarrow \mathcal{T}_\bullet(W)$
  16. ...and 1 more sections

Key Result

Theorem 1

(see Theorem T:main_thm_2) There is a natural $\mathop{\mathrm{Aut}}\nolimits_A(P)$-equivariant homotopy equivalence between $\mathcal{B}_\bullet(W)^{\Gamma\textrm{-}\mathrm{un}}$ and $\mathcal{T}_\bullet(W)$.

Theorems & Definitions (92)

  • Theorem
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Corollary 2.8
  • ...and 82 more