Representation and abelianization growth, particularly along expanding sequences
Abstract
We study two obstructions to a finite group G having a generating d-tuple whose Cayley graph has spectral gap at least ε. Lubotzky and Weiss proved that such a gap forces abk(G), the largest order of the abelianization of a subgroup of index at most k, to be exp(Oε(dk)), while de la Harpe, Robertson and Valette proved that it forces repk(G), the number of irreducible complex representations of degree at most k, to be exp(Oε(dk2)). We show that the first bound implies the second, and in fact implies it with k2 replaced by klog k. For this, we prove that every finite d-generated group satisfies the inequality repk(G)≤exp((d−1)klog k+O(dk))·abk(G), regardless of its spectral gap. The leading term (d−1)klog k is attained by powers of alternating groups, and it disappears for groups without large alternating sections. The proof reduces to counting the finite primitive subgroups of GLm(C), and we show that there are at most exp(O(log4m)) of them up to conjugacy and scalars. Wigderson asked whether k2 in the second bound can be lowered to k, so our result answers his question up to a logarithm. We do not know whether the logarithm is actually needed. For powers of alternating groups, the natural candidates, we show that a spectral gap keeps the number of factors a fixed power below the maximum allowed by d-generation, and that random generators have spectral gap tending to $0$ already for superpolynomially many factors.
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BibTeX
@misc{jezernik2026representation,
title = {{Representation and abelianization growth, particularly along expanding sequences}},
author = {Urban Jezernik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37399},
eprint = {2610.37399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37399},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Representation and abelianization growth, particularly along expanding sequences
AU - Urban Jezernik
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37399
UR - https://arxiv.org/abs/2610.37399
AB - We study two obstructions to a finite group G having a generating d-tuple whose Cayley graph has spectral gap at least ε. Lubotzky and Weiss proved that such a gap forces ab_k(G), the largest order of the abelianization of a subgroup of index at most k, to be exp(O_ε(dk)), while de la Harpe, Robertson and Valette proved that it forces rep_k(G), the number of irreducible complex representations of degree at most k, to be exp(O_ε(dk^2)). We show that the first bound implies the second, and in fact implies it with k^2 replaced by klog k. For this, we prove that every finite d-generated group satisfies the inequality rep_k(G)≤exp((d−1)klog k+O(dk))·ab_k(G), regardless of its spectral gap. The leading term (d−1)klog k is attained by powers of alternating groups, and it disappears for groups without large alternating sections. The proof reduces to counting the finite primitive subgroups of GL_m(C), and we show that there are at most exp(O(log^4m)) of them up to conjugacy and scalars. Wigderson asked whether k^2 in the second bound can be lowered to k, so our result answers his question up to a logarithm. We do not know whether the logarithm is actually needed. For powers of alternating groups, the natural candidates, we show that a spectral gap keeps the number of factors a fixed power below the maximum allowed by d-generation, and that random generators have spectral gap tending to $0$ already for superpolynomially many factors.
ER -