Related papers: Tilings and Submonoids of Metabelian Groups
In this paper we study the tensor category structure of the module category of the restricted quantum enveloping algebra associated to $\mathfrak{sl}_2$. Indecomposable decomposition of all tensor products of modules over this algebra is…
We consider semigroup algorithmic problems in finitely generated metabelian groups. Our paper focuses on three decision problems introduced by Choffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroup contain a neutral…
Let $S$ be either a free group or the fundamental group of a closed hyperbolic surface. We show that if $G$ is a finitely generated residually-$p$ group with the same pro-$p$ completion as $S$, then two-generated subgroups of $G$ are free.…
A result of D. Segal states that every complex irreducible representation of a finitely generated nilpotent group $G$ is monomial if and only if $G$ is abelian-by-finite. A conjecture of A. N. Parshin, recently proved affirmatively by I.V.…
In [BV12] we have proven that, for all hyperbolic groups and for all non-trivial free products $\Gamma$, the left-right wreath product group $G:=(Z/2Z)^{(\Gamma)} \rtimes (\Gamma \times \Gamma)$ is W$^*$-superrigid. In this paper, we extend…
We prove the first nontrivial reconstruction theorem for modular tensor categories: the category associated to any twisted Drinfeld double of any finite group, can be realised as the representation category of a completely rational…
We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free…
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite…
We discuss the set of subgroups of the automorphism group of a full shift, and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case, and free products in the group case. We also show that the…
A bi-order on a group $G$ is a total, bi-multiplication invariant order. A subset $S$ in an ordered group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to…
We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB,…
We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated…
The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of…
An infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp asked whether,…
For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every…
We show that it is impossible to algorithmically decide if the l^2-cohomology of the universal cover of a finite CW complex is trivial, even if we only consider complexes whose fundamental group is equal to the elementary amenable group…
We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply…
We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As…
This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise…
A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for…