Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quotients of Buildings by Non-uniform Lattices

Orit Sela, Mary Schaps, Uzi Vishne

TL;DR

The paper advances the understanding of quotients of Bruhat-Tits buildings by non-uniform lattices in ${\operatorname{PGL}}_d(\mathbb{F}_q((1/t)))$ for $d>2$ by constructing a canonical fundamental domain $T$ and proving that the quotient has finite volume despite not being cocompact. It develops a detailed framework for the action of stabilizers, distances, and neighbor orbits, and computes the overall co-volume via a convergent weight-sum over $T$, yielding explicit volume formulas. The core technical contribution lies in the explicit description of the quotient geometry and stabilizer structure, and in the $d=3$ case, where Hecke operators $A_1,\dots,A_2$ are analyzed to produce a rich spectrum of simultaneous eigenvectors, demonstrating non-Ramanujan behavior for non-uniform quotients. The results provide concrete tools for constructing non-cofinite quotients with controlled volume and offer insights into spectral properties of higher-rank buildings via their Hecke algebras, with implications for Ramanujan-type complexes in function-field settings.

Abstract

We consider quotients of the Bruhat-Tits building associated to the projective linear groups of dimension $d>2$ over the function field $\mathbb F_q(t)$ by a non-uniform lattice $Γ$ which is a congruence subgroup in the non-uniform lattice $ PGL_{d}(R)$, where $R=\mathbb F_q[\frac{1}{t}]$. We determine a fundamental domain and demonstrate that the quotient, while not cofinite, is at least of finite covolume. We do the case $d=3$ in considerable detail.

Quotients of Buildings by Non-uniform Lattices

TL;DR

The paper advances the understanding of quotients of Bruhat-Tits buildings by non-uniform lattices in for by constructing a canonical fundamental domain and proving that the quotient has finite volume despite not being cocompact. It develops a detailed framework for the action of stabilizers, distances, and neighbor orbits, and computes the overall co-volume via a convergent weight-sum over , yielding explicit volume formulas. The core technical contribution lies in the explicit description of the quotient geometry and stabilizer structure, and in the case, where Hecke operators are analyzed to produce a rich spectrum of simultaneous eigenvectors, demonstrating non-Ramanujan behavior for non-uniform quotients. The results provide concrete tools for constructing non-cofinite quotients with controlled volume and offer insights into spectral properties of higher-rank buildings via their Hecke algebras, with implications for Ramanujan-type complexes in function-field settings.

Abstract

We consider quotients of the Bruhat-Tits building associated to the projective linear groups of dimension over the function field by a non-uniform lattice which is a congruence subgroup in the non-uniform lattice , where . We determine a fundamental domain and demonstrate that the quotient, while not cofinite, is at least of finite covolume. We do the case in considerable detail.

Paper Structure

This paper contains 18 sections, 16 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Buildings
  3. Coxeter simplicial complexes
  4. Bruhat-Tits buildings
  5. Buildings of type $\tilde{A}_{d}$
  6. Colorings
  7. Lattices in locally compact groups
  8. A quotient of Bruhat-Tits building by a non-uniform lattice
  9. The distinguished subset $T$ of $B$
  10. The neighbors of a vertex of $B^{0}$
  11. The stabilizer of a vertex
  12. The action of the stabilizer on other vertices
  13. Distance between vertices.
  14. Disjointness of the distinguished subset $T$ of $B$ from its images under $\Gamma$
  15. The volume of the quotient of a Bruhat-Tits building by a non-uniform lattice
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G={\operatorname{PGL}}_{d}(\mathbb{F}_{q}{(\:\!\!({t})\:\!\!)})$, $\Delta = {\operatorname{PGL}}_{d}(\mathbb{F}_{q}[1/t])$ and let $B$ be a Bruhat-Tits building associated to $G$, where $d > 2$. Then for any finite index subgroup $\Gamma$ of $\Delta$, ${{\Gamma}\backslash{B}}$ is not Ramanujan

Figures (1)

  • Figure 1: Diagram of $T$

Theorems & Definitions (69)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Definition 2.9
  • ...and 59 more