Related papers: On non-isomorphic universal sofic groups

The aim of this note is to prove that there are $2^{\aleph_0}$ non-isomoprhic 2-generated just-infinite branch pro-$2$ groups.

We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ contains at least two non-quasi-isometric…

Using the model theory of metric structures, I give an alternative proof of the following result by Thomas: If the Continuum Hypothesis fails then there are power of the continuum many universal sofic groups up to isomorphism. This method…

We define a subgroup of the universal sofic group, obtained as the normaliser of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each…

It is proved that, on any Abelian group of infinite cardinality ${\bf m}$, there exist precisely $2^{2^{\bf m}}$ nonequivalent bounded Hausdorff group topologies. Under the continuum hypothesis, the number of nonequivalent compact and…

We exhibit examples of groups of intermediate growth with $2^{\aleph_0}$ ergodic, continuous, invariant random subgroups. The examples are the universal groups associated with a family of groups of intermediate growth.

Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this…

We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

We provide an infinite family of sofic one-relator groups that are not residually solvable nor residually finite. The proof is essentially different from the one in [1], as it does not require just Magnus' decompositions.

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a…

We construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq…

In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the universal sofic group. This addresses a…

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C$^\ast$-algebras which are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities…

We prove that if mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if aleph_0< lambda < 2^{aleph_0}. We also prove that if 2^{aleph_0}< mu^+< lambda =cf(lambda)< mu^{aleph_0},…

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite…

We show the set of Hopfian countable groups is a complete co-analytic subset of the standard Borel space of countable groups, resolving a question posed by Simon Thomas (https://doi.org/10.1007/s00153-017-0597-9).

A conjecture of Berkovich asserts that every non-simple finite p-group has a non-inner automorphism of order p. This conjecture is far from being proved despite the great effort devoted to it. In this paper we prove it for p-groups of…

A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one.

The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by Garc\'ia-Lucas, Margolis and del R\'io [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these…

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in 0707.0070v1 as…