[Paper Review] Topological complexity of arithmetic locally symmetric spaces

Mikołaj Frączyk, Sebastián Hurtado+1 more|arXiv (Cornell University)|Feb 28, 2022

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes that arithmetic locally symmetric spaces have topological complexity linear in volume and Betti numbers sub-linear in volume except in middle dimension. Using refined Margulis lemmas, orbital integral estimates, and Benjamini–Schramm convergence, it confirms Gelander's conjecture on uniform homotopy complexity and proves asymptotic linearity of Betti numbers with volume via $L^2$-Betti numbers.

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.

Motivation & Objective

• To establish uniform homotopy complexity for arithmetic locally symmetric spaces by bounding triangulation size relative to volume.
• To analyze the asymptotic behavior of Betti numbers in relation to volume, particularly in non-middle dimensions.
• To prove Benjamini–Schramm convergence for sequences of arithmetic lattices in semisimple Lie groups.
• To refine Margulis' lemma and orbital integral estimates for arithmetic lattices in non-compact symmetric spaces.
• To resolve the role of trace field degree in controlling the thin part and topological invariants.

Proposed method

• Develops a new arithmetic Margulis lemma to control the geometry of thin parts in arithmetic locally symmetric spaces.
• Uses estimates on orbital integrals to control the number of group elements with small displacement.
• Applies a refined version of the Margulis lemma tailored to arithmetic lattices in semisimple Lie groups.
• Employs Galois cohomology and Bruhat–Tits building theory to analyze conjugacy classes and compact cohomology classes.
• Establishes Benjamini–Schramm convergence by bounding the volume of the thin part in terms of trace field degree.
• Combines limit multiplicity results and $L^2$-Betti number theory to derive asymptotic Betti number bounds.

Experimental results

Research questions

• RQ1 Does the minimal triangulation size of arithmetic locally symmetric spaces grow linearly with volume?
• RQ2 Are Betti numbers of arithmetic locally symmetric spaces sub-linear in volume, except possibly in middle dimension?
• RQ3 Can Benjamini–Schramm convergence be established for sequences of arithmetic lattices in semisimple Lie groups with unbounded trace field degree?
• RQ4 How do orbital integrals and displacement counts constrain the geometry of thin parts in arithmetic locally symmetric spaces?
• RQ5 To what extent do compact cohomology classes in Galois cohomology affect the topology of arithmetic quotients?

Key findings

• The minimal number of tetrahedra in a triangulation of an arithmetic locally symmetric space is at most linear in the volume.
• Betti numbers grow sub-linearly with volume, except possibly in middle dimension, confirming asymptotic vanishing outside that range.
• The family of arithmetic locally symmetric spaces has uniform homotopy complexity, confirming Gelander's conjecture.
• Benjamini–Schramm convergence holds for sequences of arithmetic lattices with unbounded trace field degree, with explicit volume bounds on the thin part.
• The number of conjugacy classes with small displacement is uniformly bounded, depending only on the group type.
• For congruence arithmetic lattices, Betti numbers satisfy $b_k(M) \sim \beta_k^{(2)}(G) \cdot \mathrm{vol}(M)$, where $\beta_k^{(2)}(G)$ is the $k$-th $L^2$-Betti number of $G$.