Strong convergence of random representations of free products of finite groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912900267900928 |
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| author | Barbieri, Marco Jezernik, Urban |
| author_facet | Barbieri, Marco Jezernik, Urban |
| contents | 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 $C_2*C_3 \cong {\rm PSL}_2({\bf Z})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11905 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong convergence of random representations of free products of finite groups Barbieri, Marco Jezernik, Urban Spectral Theory Group Theory Operator Algebras Probability 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 $C_2*C_3 \cong {\rm PSL}_2({\bf Z})$. |
| title | Strong convergence of random representations of free products of finite groups |
| topic | Spectral Theory Group Theory Operator Algebras Probability |
| url | https://arxiv.org/abs/2602.11905 |