Related papers: Anti-classification results for groups acting free…

We study the first-order almost-sure theories for classes of finite structures that are specified by homomorphically forbidding a set $\mathcal{F}$ of finite structures. If $\mathcal{F}$ consists of undirected graphs, a full description of…

We construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties…

We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in…

We show that non-abelian two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded free groups. This provides an alternate proof of a theorem of A. Baudisch: that all two-generator subgroups are free or free…

In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$-module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$-module $U$ has…

We construct the first examples of genuine ergodic discrete measured groupoids that are not isomorphic to any equivalence relation or transformation groupoid. We use a construction due to B.H. Neumann of an uncountable family of pairwise…

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair…

In this work, we give two characterisations of the general linear group as a group $G$ of finite Morley rank acting on an abelian connected group $V$ of finite Morley rank definably, faithfully and irreducibly. To be more precise, we prove…

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving…

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of…

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the…

We show that the finitely generated simple left orderable groups $G_{\rho}$ constructed by the first two authors in arXiv:1807.06478 are uniformly perfect - each element in the group can be expressed as a product of three commutators of…

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We…

Let $\Aut(G)$ denote the group of (bi-)continuous automorphisms %and $\Out(G)$ the outer automorphism group of a non-Archimedean Polish group~$G$. We show that for any such $G$ with an invariant countable basis of open subgroups, the group…

We give a short geometric proof of a result of Soardi & Woess and Salvatori that a quasitransitive graph is amenable if and only if its automorphism group is amenable and unimodular. We also strengthen one direction of that result by…

For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

We introduce ideas from geometric group theory related to boundaries of groups. This is a mostly expository paper. We consider the visual boundary of a free abelian group, and show that it is an uncountable set with the trivial topology.

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the…