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Group Theory

New submissions

[ total of 7 entries: 1-7 ]
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New submissions for Fri, 12 Apr 24

[1]  arXiv:2404.07321 [pdf, ps, other]
Title: Density of growth-rates of subgroups of a free group and the non-backtracking spectrum of the configuration model
Subjects: Group Theory (math.GR); Probability (math.PR)

We prove the set of growth-rates of subgroups of a rank~$r$ free group is dense in $[1,2r-1]$. Our main technical contribution is a concentration result for the leading eigenvalue of the non-backtracking matrix in the configuration model.

[2]  arXiv:2404.07379 [pdf, other]
Title: Schur rings over ${\bf {\rm Sp}(n,2)}$ and multiplicity one subgroups
Comments: 28 pages; one figure
Subjects: Group Theory (math.GR); Combinatorics (math.CO)

We study commutative Schur rings over the symplectic groups Sp$(n,2)$ containing the class $\mathcal C$ of symplectic transvections. We find the possible partitions of $\mathcal C$ determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp$(n,2)$.

[3]  arXiv:2404.07500 [pdf, ps, other]
Title: Exact bounds for the sum of the inverse-power of element orders in non-cyclic finite groups
Authors: M. Archita
Subjects: Group Theory (math.GR)

Given a finite group $G$ of order $n.$ Denote the sum of the inverse-power of element orders in $G$ by $m(G).$ Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Suppose $G$ is a non-cyclic group of order $n$ then we show that $m(G)\geq \frac{5}{4}m(\mathbb{Z}_n).$ Our result improves the inequality $m(G)>m(\mathbb{Z}_n)$ obtained by Baniasad Azad, M., and Khorsravi B. Moreover, this bound is best as for $n=4l, l$ odd, there exists a group $G$ of order $n$ satisfying $m(G)=\frac{5}{4}m(\mathbb{Z}_n)$. Moreover, we will establish that $\frac{1}{q-1}m(G)< m(\mathbb{Z}_n)\leq \frac{4}{5}m(G),$ where $G$ is a non-cyclic group of odd order.

Cross-lists for Fri, 12 Apr 24

[4]  arXiv:2404.07350 (cross-list from math.OA) [pdf, ps, other]
Title: Random permutation matrix models for graph products
Comments: 29 pages, multiple figures. This is a subset of the previous version (v1) of arXiv:2305.19463, which we are splitting into two papers
Subjects: Operator Algebras (math.OA); Combinatorics (math.CO); Functional Analysis (math.FA); Group Theory (math.GR); Probability (math.PR)

Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain random permutation matrices, we construct random matrix models for graph independence with amalgamation over the diagonal matrices. This yields a new probabilist,ic proof that graph products of sofic groups are sofic.

Replacements for Fri, 12 Apr 24

[5]  arXiv:2402.09969 (replaced) [pdf, ps, other]
Title: On the kernel of actions on asymptotic cones
Comments: 35 pages, 4 figures. Any comments are welcome!
Subjects: Group Theory (math.GR)
[6]  arXiv:2402.16787 (replaced) [pdf, ps, other]
Title: Isoperimetric Profiles and Regular Embeddings of locally compact groups
Authors: Juan Paucar
Subjects: Group Theory (math.GR); Metric Geometry (math.MG)
[7]  arXiv:2402.19003 (replaced) [pdf, ps, other]
Title: GVZ-groups with two character degrees
Subjects: Group Theory (math.GR)
[ total of 7 entries: 1-7 ]
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