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Andreas Thom

Publications and source records attributed to Andreas Thom.

At least 19 recordsLinked to original sources
arXiv2026

Mixed identities for simple locally finite groups

A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length. We classify the infinite simple locally finite groups admitting a mixed identity: apart from an explicit list of alternating, finitary linear classical, and non-simply-laced groups of Lie type, no such group exists. For the groups in this list, we determine when mixed identities must be singular and obtain restrictions on their critical constants. Moreover, we prove that simple compact Lie groups do not admit mixed identities.

math.GR↗
arXiv2026

Nonsofic wreath products of residually finite groups

This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_{\Gamma} G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.

math.GR↗
arXiv2026

Centralizers of sofic approximations of Kazhdan groups

We prove that a Kazhdan group admitting a sofic embedding into a metric ultraproduct of symmetric groups with a centralizer that acts ergodically on the associated Loeb probability space is locally embeddable in finite groups (LEF). In particular, every finitely presented Kazhdan group admitting such an embedding is residually finite. The main technical theorem says that the centralizer of a sofic embedding of a Kazhdan group is itself a metric ultraproduct of permutation groups.

math.GR↗
arXiv2026

Trace-norm rigidity for reduced products of unitary groups and matrix algebras

We study homomorphisms, with a focus on isomorphisms, between the tracial metric reduced products of finite dimensional unitary groups and of matrix algebras. A variant of Ulam stability for unitary groups and a classification of the almost surjective continuous homomorphisms between finite dimensional unitary groups are proved and then used to show that all isomorphisms of product form of these tracial reduced products are induced by almost permutations of the coordinates and coordinatewise application of automorphisms. We prove coordinate recognition for these reduced products and obtain under set theoretic assumptions rigidity and classification results for their full automorphism groups. For tracial reduced matrix algebras we obtain such rigidity result in the more general context of center-preserving $*$-homomorphisms.

math.OA↗
arXiv2026

$L^2$-cohomology and deformations of the left regular representation

We study deformations of unitary representations $\pi\colon G\to U(H)$ whose coefficients lie in the Hilbert-Schmidt ideal $HS(H)\subset B(H)$. Interesting applications arise for the left-regular representation of surface groups and, more generally, cocompact lattices in the automorphism group of a Fuchsian building of conformal dimension $<2$.

math.OA↗
arXiv2026

On the Howe--Moore property for automorphism groups of buildings

Let $G$ be a closed type-preserving subgroup of the automorphism group of a thick locally finite building $X$ of finite rank, and assume that $G$ acts Weyl-transitively. We prove that every unitary representation of $G$ is mixing, unless its restriction to a parabolic subgroup of minimal non-spherical type is amenable in the sense of Bekka. It follows that every unitary representation of $G$ that is weakly contained in the regular representation, is mixing. In case $X$ is of minimal non-spherical type and its thickness satisfies some modest lower bound, we deduce that $G$ has the Howe--Moore property provided its only compact quotient is trivial. We also obtain results on rigidity of invariant random subgroups for Kac--Moody lattices of compact hyperbolic type, yielding examples of infinite finitely presented Kazhdan groups with exactly two ergodic invariant random subgroups.

math.GR↗
arXiv2026

The hyperfinite II$_1$-factor is Ulam stable

We prove Ulam stability of the hyperfinite II$_1$-factor with respect to the trace norm on the operator-norm unit ball. More precisely, every sufficiently additive, multiplicative, unital, $*$-preserving map from the hyperfinite II$_1$-factor-factor into a II$_1$-factor-factor von Neumann algebra is uniformly close, after passing to a small amplification of the target, to a genuine unital $*$-homomorphism. As a key finite-dimensional ingredient, we establish a dimension-free stability theorem for matrix algebras in the same trace-norm setting. As an application, we show that the hyperfinite II$_1$-factor is isolated among II$_1$-factors with respect to sufficiently accurate approximate $*$-isomorphisms.

math.OA↗
arXiv2026

Ulam stability for classes of nuclear C*-algebras

We study Ulam stability for approximate *-homomorphisms of C*-algebras. We prove stability results for several classes of nuclear C*-algebras with respect to von Neumann algebra targets, including abelian C*-algebras and large classes arising in the Elliott classification program. We also discuss permanence properties, counterexamples, and related stability phenomena. As applications, we obtain rigidity and independence results for corona algebras.

math.OA↗
arXiv2026

On prime endomorphisms of the free group of rank two

We study prime endomorphisms of the free group $F_2$. The main results provide lower and upper bounds for the logarithmic density $\delta_{\mathrm{np}}$ of non-prime endomorphisms: $$ \frac34 \leq \delta_{\mathrm{np}}\leq \frac{27}{28}. $$ Equivalently, we prove corresponding lower and upper bounds for the exponential decay of the probability that a random endomorphism is non-prime, aligning with a heuristic argument of Ian Agol.

math.GR↗
arXiv2026

Sublinear growth of 1-cocycles and uniform convexity

Let G be a finitely generated group, let $\pi \colon G \to {\rm GL}(E)$ be a uniformly bounded $c_0$-representation on a superreflexive Banach space $E$, and let $b \colon G \to E$ be a $1$-cocycle for $\pi$. Then $b$ has sublinear growth with respect to the word length. As a corollary we obtain the corresponding Hilbert space statement for strongly mixing unitary representations.

math.GR↗
arXiv2026

Cubic maps from the group of order $3$

The purpose of this note is to classify unital cubic maps from the cyclic group of order $3$ into an arbitrary non-abelian group. We show that the universal group admitting a unital cubic map from the cyclic group of order $3$ is infinite, give a concrete presentation and provide an infinite representation of it in ${\rm PSL}_3(\mathbb C)$, whose image is an arithmetic lattice commensurable with ${\rm PSL}_3(\mathbb Z[\omega])$, where $\omega$ is a primitive cube root of unity. As a consequence we obtain the existence of finite nilpotent groups of arbitrarily large nilpotency class admitting a unital cubic map from $C_3$ whose image generates the group.

math.GR↗
arXiv2025

Remarks on approximability and stability for groups

In this paper, we provide several instances in which interesting approximation and stability properties are inherited by quotients with respect to finitely generated normal subgroups or, more strongly, normal subgroups with Kazhdan's property (T). Applications arise when these observations are combined with variations of the Rips construction due to Wise and Belegradek--Osin.

math.GR↗
arXiv2025

On planar sections of the dodecahedron

In the analysis of three-dimensional biological microstructures such as organoids, microscopy frequently yields two-dimensional optical sections without access to their orientation. Motivated by the question of whether such random planar sections determine the underlying three-dimensional structure, we investigate a discrete analogue in which the ambient structure is the vertex set of a Platonic solid and the observed data are congruence classes of planar intersections. For the regular dodecahedron with vertex set $V$, we define the planar statistic of a subset $X\subseteq V$ of vertices as the distribution of isometry types of inclusions $\Pi\cap X \subseteq \Pi \cap V \subseteq V$, and ask whether this statistic determines $X$ up to isometry. We show that this is not the case: there exist two non-isometric $7$-element subsets with identical planar statistics. As a consequence, there exist two polytopes in $\mathbb R^3$, whose distribution of isometry classes of two-dimensional intersections is identical, while the polytopes are not themselves isometric. This result is an analogue of classical non-uniqueness phenomena in geometric tomography.

math.MG↗
arXiv2025

The anabelian restricted Burnside problem

Let $n,d \in \mathbb N$ and $w \in \mathbb F_n$ be non-trivial. We prove that the relatively free group of rank $d$ in the variety defined by the group law $w$ has a largest anabelian finite quotient and estimate its size. Here, a finite group is called anabelian if it has only non-abelian composition factors. The estimate is based on explicit bounds for the length of laws for finite simple groups obtained by Bradford and the author and on recent work by Fumagalli--Leinen--Puglisi.

math.GR↗
arXiv2025

On finite approximations of transitive graphs

In this note we answer a question of Johannes Carmesin, which was circulated at the Oberwolfach Workshop on "Graph Theory" in January 2025. We provide a unimodular, locally finite, and vertex-transitive graph without any perfect finite $r$-local model for $r \in \mathbb N$ large enough.

math.CO↗