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Strong convergence of random representations of free products of finite groups

Marco Barbieri, Urban Jezernik

Abstract

We extend the polynomial method of Chen--Garza-Vargas--Tropp--van Handel and Magee--Puder--van Handel for operator-norm bounds in random permutation models to the setting where torsion is present. The main new feature is that asymptotic expansion of traces naturally involves fractional powers of $N$ rather than an ordinary Laurent series. We formulate fractional-power analogues of the method's key hypotheses and prove they lead to strong convergence. We verify these analogues for free products of finite groups $Γ=G_1*\cdots*G_m$. Concretely, for a uniformly random $φ_N\in{\rm hom}(Γ,{\rm Sym}(N))$, set $π_N = {\rm std} \circ φ_N$, where ${\rm std}$ denotes the standard $(N-1)$-dimensional representation of ${\rm Sym}(N)$ (the permutation representation with the trivial subrepresentation removed). We deduce strong convergence of $π_N$ to the left regular representation of $Γ$. As applications, we obtain asymptotically sharp spectral gaps for the associated random Schreier graphs, including almost Ramanujan behavior for $C_2*C_2*C_2$ and an explicit non-Ramanujan limiting spectral radius for $C_2*C_3 \cong {\rm PSL}_2({\bf Z})$.

Strong convergence of random representations of free products of finite groups

Abstract

We extend the polynomial method of Chen--Garza-Vargas--Tropp--van Handel and Magee--Puder--van Handel for operator-norm bounds in random permutation models to the setting where torsion is present. The main new feature is that asymptotic expansion of traces naturally involves fractional powers of rather than an ordinary Laurent series. We formulate fractional-power analogues of the method's key hypotheses and prove they lead to strong convergence. We verify these analogues for free products of finite groups . Concretely, for a uniformly random , set , where denotes the standard -dimensional representation of (the permutation representation with the trivial subrepresentation removed). We deduce strong convergence of to the left regular representation of . As applications, we obtain asymptotically sharp spectral gaps for the associated random Schreier graphs, including almost Ramanujan behavior for and an explicit non-Ramanujan limiting spectral radius for .
Paper Structure (29 sections, 39 theorems, 235 equations, 1 figure)

This paper contains 29 sections, 39 theorems, 235 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Expander graphs
  3. The polynomial method
  4. Expansion in random graphs with torsion restrictions
  5. Methods
  6. Reader's guide
  7. Assumptions imply strong convergence
  8. Asymptotics for exponentials of polynomials
  9. Quantitative asymptotics in terms of $r_n$
  10. Approximating $r_n$
  11. Expanding $r_n$
  12. Approximating $r_n$ by $\rho_n(s)$
  13. Replacing $r_n$ by $\rho_n(s)$ in \ref{['thm:quantitative_muller_rn']}
  14. Quantitative asymptotics in terms of $n$
  15. Fractional expansions for the number of group actions
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a free product of finite groups as in eq:Gamma_free_product_of_Gi. Then the random representations $\pi_N = \mathop{\mathrm{\mathrm{std}}}\nolimits \circ \phi_N$ strongly converge to the left regular representation of $\Gamma$.

Figures (1)

  • Figure 1: The spectral measure of the adjacency operator of $C_2*C_3$. The support of the measure consists of the union of two intervals with boundary points at $(1 \pm \sqrt{13 + 8 \sqrt{2}})/2$ and $(1 \pm \sqrt{13 - 8 \sqrt{2}})/2$, together with discrete atoms at $-2$ and $0$, each with mass $1/6$. These atoms correspond to torsion-induced $\ell^2$-eigenfunctions $f$ of the adjacency operator $\mathcal{A} f(g)=f(xg)+f(yg)+f(y^2g)$. Imposing that $f$ has sum zero on each $C_3$-triangle $\{g,yg,y^2g\}$ reduces the eigenvalue equation $\mathcal{A} f = \lambda f$ to $f(xg)=(\lambda+1)f(g)$, and the relation $x^2=1$ forces $\lambda+1 = \pm1$.

Theorems & Definitions (77)

  • Example
  • Theorem 1.1: strong convergence for free products of finite groups
  • Example
  • Example
  • Theorem 1.2
  • Remark
  • Lemma 2.1: Lemma 4.2 in MageeSalle2024
  • Proposition 2.2: master inequality
  • proof
  • Proposition 2.3
  • ...and 67 more