Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.03960

Invariant random subgroups of the finitary special linear group over a finite field

Noam Atar·Arie Levit
View original source ↗Read source document ↗Export metadata ↓

Abstract

An invariant random subgroup of a discrete group is a conjugation-invariant probability measure on the space of its subgroups. We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field. These are shown to be stabilizers of random tuples of vectors and functionals. The classification uses the character theory of the finitary special linear group established by Skudlarek and the theory of the invariant random subgroups of the finitary permutation group established by Vershik. In addition, we classify the invariant probability measures for several natural actions of these finitary groups.

Explore related subjects

math.GRmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Noam Atar, Arie Levit. 2026-11-02. Invariant random subgroups of the finitary special linear group over a finite field. https://arxiv.org/abs/2611.03960

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On a Completion of Cohomological Functors Generalising Tate Cohomology I

Tate cohomology has been generalised by several authors using different constructions that have applications in group theory, ring theory and homotopical algebra. Therefore, there is a need for a uniform account that explains why their underlying approaches all lead to the same conclusions. The key notion in such a uniform theory is a specific completion of cohomological functors that is constructed under mild assumptions. This completion takes Tate cohomology to settings where it has never been introduced such as in condensed mathematics. Through the latter, one can define Tate cohomology for any $T1$ topological group.

math.GR↗
arXiv2026

$L^2$-Betti numbers of Dehn fillings

We initiate the study of the $L^2$-Betti numbers of group-theoretic Dehn fillings. For a broad class of virtually special groups $G$, we prove that the $L^2$-Betti numbers of sufficiently deep Dehn fillings $\overline{G}$ are equal to those of $G$. As applications, we verify the Singer Conjecture for certain Einstein manifolds, establish a virtual fibering criterion for $\overline{G}$, obtain bounds on deficiency of $\overline{G}$, and provide new examples of hyperbolic groups with exotic subgroups that arise as Dehn fillings of any cusped arithmetic hyperbolic manifold of dimension at least four.

math.GR↗
arXiv2026

On Zappa-Szép products of two semidihedral groups

Let $n, m \ge 4$. We study the Zappa--Szép products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong SD_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong SD_{2^m}$, together with the fixed generators $x, y, z, w$, and we describe the parameter tuples which encode the mixed commutators of these generators, according to the cores of $\langle x\rangle$ and $\langle z\rangle$ in~$G$. First, when both $\langle x\rangle$ and $\langle z\rangle$ are normal in~$G$, we show that the admissible parameter tuples are exactly the solutions of an explicit system of six polynomial congruences. Second, when the cores $\langle x\rangle_G$ and $\langle z\rangle_G$ are arbitrary subgroups of $\langle x\rangle$ and $\langle z\rangle$, under the simplifying assumption $[x, z] = 1$, and for generating systems for which all mixed commutators lie in $\langle x\rangle\langle z\rangle$, we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu--Yu classification~\cite{HuYu2025} for dihedral groups $D_{2n}$ and $D_{2m}$ with $n, m$ odd. In contrast with that case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, so that the semidihedral setting contains additional families of exact products.

math.GR↗