Related papers: On the Cantor-Bendixson rank of metabelian groups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse…

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3.

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to…

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for $R=\mathbb Q$ or $R=\mathbb Z/n$ and a finitely presented metabelian group $G$, the natural map from $G$ to its…

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$,…

Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does…

For a finite group $G$, we associate the quantity $\beta(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second…

We introduce the concept of a generic Euclidean triangle $\tau$ and study the group $G_\tau$ generated by the reflection across the edges of $\tau$. In particular, we prove that the subgroup $T_\tau$ of all translations in $G_\tau$ is free…

We present a new method for showing that groups are virtually special. This is done by considering finite quotients and linear characters. We use this to show that an infinite family of groups, related to Bestvina-Brady groups and…

In previous work with Harman, we introduced a new class of representations for an oligomorphic group $G$, depending on an auxiliary piece of data called a measure. In this paper, we look at this theory when $G$ is the symmetry group of the…

We investigate the algebraic K- and L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.

We examine dimensional types of scattered $P$-spaces of weight $\omega_1$. Such spaces can be embedded into $\omega_2$. There are established similarities between dimensional types of scattered separable metric spaces and dimensional types…

We consider groups of finite Morley rank with solvable local subgroups of even and mixed types. We also consider miscellaneous aspects of small groups of finite Morley rank of odd type.

We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real…

In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen's polytope algebra, with the universal property that continuous valuations on convex bodies…

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We…

We present results about the Cantor-Bendixson index of some subspaces of a uniform family F of finite subsets of natural numbers with respect to the lexicographic order topology. As a corollary of our results we get that for any…

We classify rank one 2-representations of SL2, GL2 and SO3 web categories. The classification is inspired by similar results about quantum groups, given by reducing the problem to the classification of bilinear and trilinear forms, and is…

Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first…

In the paper we study irreducible representations of some nilpotent groups of finite abelian total rank. The main result of the paper states that if a torsion-free minimax group $G$ of nilpotency class 2 admits a faithful irreducible…