Topological complexity of arithmetic locally symmetric spaces
Number Theory
2026-02-03 v2 Group Theory
Geometric Topology
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.
Keywords
Cite
@article{arxiv.2202.13619,
title = {Topological complexity of arithmetic locally symmetric spaces},
author = {Mikołaj Frączyk and Sebastian Hurtado and Jean Raimbault},
journal= {arXiv preprint arXiv:2202.13619},
year = {2026}
}
Comments
Previous version split in half, this is the first part and the second will appear shortly on arxiv