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Andreas Thom

Publications and source records attributed to Andreas Thom.

At least 37 records · Page 2Linked to original sources
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 ${\rm PSL}_3(\mathbb Z[ω])$, where $ω$ 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↗
arXiv2026

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 $Π\cap X \subseteq Π\cap V \subseteq V$, 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

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

Amenability and skew-amenability of actions of topological groups

We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes to a closed subgroup. Other applications include the understanding of the universal minimal flow of various non-amenable groups.

math.GR↗
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

Quadratic maps between non-abelian groups

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a $1\%$-inverse theorem for the $U^2$-norm and relating it to stability questions for almost representations. In this article, we take a step toward an inverse theory for higher-order uniformity norms of matrix-valued functions on arbitrary groups by examining the $99\%$ regime for the $U^k$-norm on perfect groups of bounded commutator width. This analysis prompts a classification of Leibman's quadratic maps between non-abelian groups. Our principal contribution is a complete description of these maps via an explicit universal construction. From this classification we deduce several applications: A full classification of quadratic maps on arbitrary abelian groups; a proof that no nontrivial polynomial maps of degree greater than one exist on perfect groups; stability results for approximate polynomial maps.

math.GR↗
arXiv2025

Mixed identities for oligomorphic automorphism groups

We study mixed identities for oligomorphic automorphism groups of countable relational structures. Our main result gives sufficient conditions for such a group to not admit a mixed identity without particular constants. We study numerous examples and prove in many cases that there cannot be a non-singular mixed identity.

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↗
arXiv2024

Amenability for unitary groups of C*-algebras

In this note we state a conjecture that characterizes unital C*-algebras for which the unitary group is amenable as a topological group in the norm topology. We prove the conjecture for simple, separable, stably finite, unital, $\mathcal Z$-stable, UCT C*-algebras with torsionfree K_0 using the progress on the Elliott classification program for nuclear C*-algebras as well as Pestov's study of amenability of gauge groups. Based on work of Kirchberg, we provide a counterexample to a question of Ng, who proposed a different characterization in earlier work.

math.OA↗
arXiv2024

High-dimensional expansion and soficity of groups

For $d \geq 4$ and $p$ a sufficiently large prime, we construct a lattice $Γ\leq {\rm PSp}_{2d}(\mathbb Q_p),$ such that its universal central extension cannot be sofic if $Γ$ satisfies some weak form of stability in permutations. In the proof, we make use of high-dimensional expansion phenomena and, extending results of Lubotzky, we construct new examples of cosystolic expanders over arbitrary finite abelian groups.

math.GR↗
arXiv2023

Non-singular word maps for linear groups

We study the word image of words with constants in ${\rm GL}(V)$ and show that it is large provided the word satisfies some natural conditions on its length and its critical constants. There are various consequences: We prove that for every $l \geq 1$, there are only finitely many pairs $(n,q)$ such that the length of the shortest non-singular mixed identity ${\rm PSL}_n(q)$ is bounded by $l$. We generalize the Hull--Osin dichotomy for highly transitive permutation groups to linear groups over finite fields. Finally, we show that the rank limit of ${\rm GL}_n(q)$ for $q$ fixed and $n \to \infty$ is mixed identity free.

math.GR↗
arXiv2023

Common transversals for coset spaces of compact groups

Let $G$ be a Polish group and let $H \leq G$ be a compact subgroup. We prove that there exists a Borel set $T \subset G$ which is simultaneously a complete set of coset representatives of left and right cosets, provided that a certain index condition is satisfied. Moreover, we prove that this index condition holds provided that $G$ is locally compact and $G/G^\circ$ is compact or $H$ is a compact Lie group. This generalizes a result which is known for discrete groups under various finiteness assumptions, but is known to fail for general inclusions of infinite groups. As an application, we prove that Bohr closed subgroups of countable, discrete groups admit common transversals.

math.GR↗
arXiv2023

About discrete subgroups of full groups of measure preserving equivalence relations

In this note we study countable subgroups of the full group of a measure preserving equivalence relation. We provide various constraints on the group structure, the nature of the action, and on the measure of fixed point sets, that imply that the subgroup topology is not discrete. We mention various conjectures about discrete subgroups of full groups.

math.GR↗
arXiv2023

On the length of non-solutions to equations with constants in some linear groups

We show that for any finite-rank free group $Γ$, any word-equation in one variable of length $n$ with constants in $Γ$ fails to be satisfied by some element of $Γ$ of word-length $O(\log (n))$. By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group $Γ$. Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including $\mathrm{PSL}_d(\mathbb{Z})$ for all $d \geq 2$, and the fundamental groups of all closed hyperbolic surfaces and $3$-manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group $Γ$ and a sequence of word-equations with constants in $Γ$ for which every non-solution in $Γ$ is of word-length strictly greater than logarithmic.

math.GR↗
arXiv2023

Some thoughts and experiments on Bergman's compact amalgamation problem

We study the question whether copies of $S^1$ in $\mathrm{SU}(3)$ can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. Considerable computational experiments suggest that the answer is positive in this case. We obtain a positive answer for a relaxed problem using theoretical considerations.

math.GR↗
arXiv2023

The length of mixed identities for finite groups

We prove that there exists a constant $c>0$ such that any finite group having no non-trivial mixed identity of length $\leq c$ is an almost simple group with a simple group of Lie type as its socle. Starting the study of mixed identities for almost simple groups, we obtain results for groups with socle ${\rm PSL}_n(q)$, ${\rm PSp}_{2m}(q)$, ${\rm P Ω}_{2m-1}^\circ(q)$, and ${\rm PSU}_n(q)$ for a prime power $q$. For such groups, we will prove rank-independent bounds for the length of a shortest non-trivial mixed identity, depending only on the field size $q$.

math.GR↗