Related papers: Which groups are amenable to proving exponent two …

We prove that there is a second countable locally compact group that does not embed as a closed subgroup in any compactly generated locally compact group, and discuss various related embedding and non-embedding results.

In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and…

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a…

The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the…

This is the second paper in a series on enumerative invariants counting self-dual objects in self-dual categories, and is a sequal to (arXiv:2302.00038). Ordinary enumerative invariants in abelian categories can be seen as invariants for…

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\geq 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to…

We advance the Cohn-Umans framework for developing fast matrix multiplication algorithms. We introduce, analyze, and search for a new subclass of strong uniquely solvable puzzles (SUSP), which we call simplifiable SUSPs. We show that these…

We determine the irreducible 2-modular representations of the symplectic group $G=Sp_{2n}(2)$ whose restriction to every abelian subgroup has a trivial constituent. A similar result is obtained for maximal tori of $G$. There is significant…

Let $\mathbb F$ be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in ${\rm Sp}(2n,\mathbb F)$. We show that ultralimits of maximal representations in…

Let $U$ be an algebraic subgroup of the group of $n\times n$ upper-triangular matrices with units on the diagonal over a finite field of large enough characteristic, and $\mathfrak{n}$ be the Lie algebra of $U$. The main tool in…

We calculate the mod-two cohomology of all alternating groups together, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show…

We show that a large family of groups without non-abelian free subgroups satisfy the following strengthening of non-amenability: they each have a rich supply of irreducible representations defining exotic C*-algebras. The construction is…

We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as \emph{normal} sets. We show that for any $\epsilon>0$, there exists $n_0$ such that if $n>n_0$ and $A$ is a normal subset of…

The symplectic group branching algebra, B, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp(2n-2,C) in each irreducible representation of Sp(2n,C). By describing on B an ASL structure, we…

A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|f\sigma \cap f\tau|$ is even for any non-adjacent $k$-faces…

It is conjectured that irreducible representations of symmetric groups have no non-trivial self-extension over fields of odd characteristic. We improve on partial results showing evidence of this conjecture.

Let $\Gamma$ be a countable abelian group. An (abstract) $\Gamma$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\Gamma$ - is said to be a Conze-Lesigne system if…

We study finite semigroups of $n \times n$ matrices with rational entries. Such semigroups provide a rich generalization of transition monoids of unambiguous (and, in particular, deterministic) finite automata. In this paper we determine…

We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences…

The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture…