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Topological complexity of arithmetic locally symmetric spaces

Mikołaj Frączyk, Sebastian Hurtado, Jean Raimbault

Abstract

We prove that any arithmetic locally symmetric space is homotopy equivalent to a simplicial complex where the number of simplices is bounded linearly in the volume of the space. This settles a well-known conjecture of Gelander. The main technical ingredient, which is of independent interest, is a strengthened version of the Margulis' collar lemma for arithmetic locally symmetric spaces based on the height gap theorem of Breuillard, in which the Margulis constant is made linear in the degree of the trace field of the lattice.

Topological complexity of arithmetic locally symmetric spaces

Abstract

We prove that any arithmetic locally symmetric space is homotopy equivalent to a simplicial complex where the number of simplices is bounded linearly in the volume of the space. This settles a well-known conjecture of Gelander. The main technical ingredient, which is of independent interest, is a strengthened version of the Margulis' collar lemma for arithmetic locally symmetric spaces based on the height gap theorem of Breuillard, in which the Margulis constant is made linear in the degree of the trace field of the lattice.
Paper Structure (16 sections, 23 theorems, 66 equations)

This paper contains 16 sections, 23 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Topological complexity
  3. Arithmetic Margulis lemma.
  4. Preliminaries on arithmetic lattices
  5. Arithmetic lattices
  6. Weil height and minimal displacement
  7. Dobrowolski's bounds and injectivity radius
  8. Arithmetic collar lemma
  9. Uniform growth of orbital integrals
  10. Orbital integrals
  11. Uniform growth
  12. Volume of the thin part
  13. Proof of Theorem \ref{['Main_thinpart']}
  14. Homotopy type
  15. Triangulations
  16. ...and 1 more sections

Key Result

Theorem A

There are constants $A,B$ dependent only on the symmetric space $X$, such that every closed arithmetic manifold $M=\Gamma\backslash X$ is homotopy equivalent to a simplicial complex $\mathcal{N}$ with at most $A \operatorname{vol}(M)$ simplices, where every vertex is incident to at most $B$ simplice

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 31 more