arXiv subjects
Publications and source records attributed to Vadim Alekseev.
We prove that the relative commutant of a finite unitary tuple in a tracial matrix ultraproduct is internal whenever its conjugation operator has a spectral gap off the commutant. This implies the existence of a non-hyperlinear group.
We study word maps with constants on quasisimple algebraic groups over a local field $L$. We prove that, for such a group $G=\mathbf{G}(L)$ and a word $w\in (G\ast\mathbf{F}_r)\setminus G$, either $w$ has a so-called Tomanov-small critical constant or the minimal dimension of the word image $w(G^r)\subseteq G$ of such a word is bounded from below by a function $c(G)>1$. We compute the optimal value of $c(G)$ for most quasisimple linear algebraic 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.
We show that there are $2^{\aleph_0}$ non-isomorphic universal sofic groups. This proves a conjecture of Simon Thomas.
We prove an asymptotic equidistribution result for word values for words with constants in the symmetric group. We also speculate about simultaneous asymptotic equidistribution results for values of $d$-tuples of elements of $\mathbb F_d$.
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.
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.
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.
We introduce the notion of a ``sofic $\mathcal{C}$-action'' of one group on another by automorphisms, for $\mathcal{C}$ a class of groups. We show that if $\mathcal{C}$ is the class of (i) sofic, (ii) hyperlinear, (iii) linear sofic or (iv) weakly sofic groups, then the class $\mathcal{C}$ is closed under taking semidirect products with sofic $\mathcal{C}$-action. We use this to construct a wide variety of new examples of groups in the classes (i)-(iv), many of them arising as ``halo products'' in the sense of Genevois-Tessera. We have a parallel set of results producing new examples of semidirect products which are locally embeddable into finite groups. Our framework also unifies existing results in the literature, due to Hayes-Sale; Brude-Sasyk and Gao-Kunnawalkam Elayavalli-Patchell.
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.
We show that any isometric action of a residually finite group admits approximate local finite models. As a consequence, if $G$ is residually finite, every isometric $G$-action embeds isometrically into a metric ultraproduct of finite isometric $G$-actions.
We prove that every sofic approximation of a property (T) group is approximately isomorphic to one having geometric property (T), and more generally, a box space of graphs which has boundary geometric property (T) is approximately isomorphic to one having geometric property (T). We also prove that a sequence of bounded degree graphs is approximately isomorphic to a disjoint union of expanders if and only if the Laplacian has spectral gap in the ultraproduct. Finally, we prove a local geometric criterion for geometric property (T) in the spirit of Żuk's criterion for property (T) for 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.
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.
We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, Koopman and groupoid-left-regular representations. Besides that, we unify several recent results about paradoxicality in semigroups and groupoids, relating embeddings of Thompson's group V into full groups of ample étale groupoids.
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.
By classifying $S$-maximal amenable subgroups of algebraic groups over a global field of characteristic zero, we obtain a complete classification of maximal amenable subgroups up to commensurability in the respective arithmetic groups. Futhermore, we prove that these commensurably maximal amenable subgroups are singular and therefore give rise to maximal amenable von Neumann subalgebras.
We show that if a group G is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing G. The proofs are elementary and make use of free probability theory. In addition, we clarify the situation for C*-algebras.