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Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$

Piotr Mizerka

TL;DR

The paper extends the SOS-based spectral-gap method from degree-zero to degree-one cohomological Laplacians, enabling a constructive lower bound for the first Laplacian $\Delta_1$ of groups like $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{SAut}(F_n)$. It develops Fox-derivative models of $\Delta_1$, decomposes it into square, adjacent, and opposite parts, and proves a robust symmetrization framework that propagates gaps from small $n$ to all larger $n$. The main result shows $\Delta_1-0.217(n-2)I_{|\mathcal{S}_n|}$ is a sum of squares for $\lambda=0.217(n-2)$, yielding an explicit spectral-gap certificate and thereby reaffirming property (T) for $n\ge 3$ (and its extension to the $\mathrm{SAut}(F_n)$ setting with caveats). The paper also provides practical replication details, including code and methodology, to certify the SOS certificates for $\mathrm{SL}_3(\mathbb{Z})$ and discusses the impact of these bounds as explicit, scalable indicators of cohomological vanishing and Kazhdan constants.

Abstract

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\operatorname{SAut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$. In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for $\operatorname{SL}_n(\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for $\operatorname{SL}_n(\mathbb{Z})$ whenever $n\geq 3$.

Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$

TL;DR

The paper extends the SOS-based spectral-gap method from degree-zero to degree-one cohomological Laplacians, enabling a constructive lower bound for the first Laplacian of groups like and . It develops Fox-derivative models of , decomposes it into square, adjacent, and opposite parts, and proves a robust symmetrization framework that propagates gaps from small to all larger . The main result shows is a sum of squares for , yielding an explicit spectral-gap certificate and thereby reaffirming property (T) for (and its extension to the setting with caveats). The paper also provides practical replication details, including code and methodology, to certify the SOS certificates for and discusses the impact of these bounds as explicit, scalable indicators of cohomological vanishing and Kazhdan constants.

Abstract

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for and . In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for whenever .
Paper Structure (11 sections, 10 theorems, 47 equations, 2 figures)

This paper contains 11 sections, 10 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fox derivatives and sums of squares
  3. Presentations of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$
  4. Decomposition into square, adjacent and opposite part
  5. Sum of squares decomposition for square and opposite part
  6. Symmetrization of the adjacent part
  7. General setting for symmetrization
  8. Invariance of Laplacians
  9. Adjacent part symmetrizes well
  10. Application to $G_n=\operatorname{SL}_n(\mathbb{Z})$
  11. Replication details for $\operatorname{SL}_3(\mathbb{Z})$

Key Result

Theorem 1.1

Suppose $\operatorname{Adj}_m-\lambda I_{|\altmathcal{S}_m|}$ is a sum of squares in $\mathbb{M}_{|\altmathcal{S}_m|\times |\altmathcal{S}_m|}(\mathbb{R} G_m)$. Then is a sum of squares in $\mathbb{M}_{|\altmathcal{S}_n|\times |\altmathcal{S}_n|}(\mathbb{R} G_n)$ for any $n\geq m$.

Figures (2)

  • Figure 1: The simplex $C_3$ with the assignments of the generators of $\operatorname{SL}_3(\mathbb{Z})$ to its edges given by $\phi$.
  • Figure 2: The simplex $C_4$ with the assignments of the generators of $\operatorname{SAut}(F_4)$ to its edges given by $\phi$.

Theorems & Definitions (22)

  • Theorem 1.1: cf. \ref{['corollary:main_ind']}
  • Corollary 1.2: cf. \ref{['corollary:main']}
  • Definition 2.1
  • Remark 2.2
  • Example 4.1
  • Example 4.2
  • Lemma 4.3
  • proof
  • Lemma 5.1
  • proof
  • ...and 12 more