Related papers: Remarks Concerning Lubotzky's Filtration

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include…

A theorem of Glasner from 1979 shows that if $Y \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $\epsilon > 0$ there exists an integer $n$ such that $nY$ is $\epsilon$-dense. This has been extended in various works by…

There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt…

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic…

We prove that every {finitely generated residually finite}-by-sofic group satisfies Kaplansky's direct and stable finiteness conjectures with respect to all noetherian rings. We use this result to provide countably many new examples of…

Dynamical systems whose linearizations along trajectories are positive in the sense that they infinitesimally contract a smooth cone field are called differentially positive. The property can be thought of as a generalization of…

Continual Lie algebras are infinite-dimensional generalizations of Lie algebras with discrete root system by considering continual root systems. In this paper we establish the general relation between chain complexes and continual Lie…

For a formation $\mathfrak{F}$ of finite groups consider a graph whose vertices are elements of a finite group and two vertices are connected by an edge if and only if they generates non-$\mathfrak{F}$-group as elements of a group. A…

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of…

We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\em binary subgroups}, $B(\Sigma,\mu)<G_1\times\dots\times G_m$. These full subdirect products require…

We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of…

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be…

We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These…

We study an effective gauge theory whose gauge group is a semidirect product $G = G_c \rtimes \mathit{\Gamma}$ with $G_c$ and $\mathit{\Gamma}$ being a connected Lie group and a finite group, respectively. The semidirect product is defined…

Given a $p$-adic group $G$ equipped with an action of a finite group $\Gamma\subset\mathrm{Aut}_F(\mathbf{G})$, and a reductive fixed-point subgroup $G^\Gamma$, we establish a relationship between constructions of types for these two groups…

We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of…

These notes expand upon our lectures on {\em profinite rigidity} at the international colloquium on randomness, geometry and dynamics, organised by TIFR Mumbai at IISER Pune in January 2024. We are interested in the extent to which groups…

By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we…

Let $(\pi, \mathcal{H})$ be a strongly continuous unitary representation of a 1-connected Lie group $G$ such that the Lie algebra $\mathfrak{g}$ of $G$ is generated by the positive cone $C_\pi := \{x \in \mathfrak{g} : -i\partial \pi(x)…