Related papers: Normal Subgroup Theorem for groups acting on $\til…

We study the class of finite groups $G$ satisfying $\Phi (G/N)= \Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe…

The problem of equivariant rigidity is the $\Gamma$-homeomorphism classification of $\Gamma$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $\Gamma$. In other words, this is the…

It is shown that most lattices $\Gamma$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ possess a fundamental domain $F$ for the action of $\Gamma$ on $\mathbb{R}^2$, respectively $\mathbb{R}^3$, having more symmetries than the point group…

Let $G$ be a connected and simply connected semisimple algebraic group over $\Bbb Q$ and let $\Gamma\subset G(\Bbb Q)$ be an arithmetic subgroup. Let $K_\infty\subset G(\Bbb R)$ be a maximal compact subgroup and let $d$ be the dimension of…

Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $\Gamma$ with Sylow $2$-subgroup $\Gamma_2$ and an infinite family of pairwise non-conjugate…

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a…

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception (only in…

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem: Let C be a large homogeneous model of a stable diagram D. Let p, q in S_D(A), where p is…

We say that a finitely generated group $\Gamma$ is self-simulable if every effectively closed action of $\Gamma$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $\Gamma$-subshift of finite type.…

Let $X$ be a locally finite irreducible affine building of dimension $\geq 2$ and $\Gamma \leq \mathrm{Aut}(X)$ be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is $\Gamma$ linear?…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…

The end compactification |\Gamma| of the locally finite graph \Gamma is the union of the graph and its ends, endowed with a suitable topology. We show that \pi_1(|\Gamma|) embeds into a nonstandard free group with hyperfinitely many…

We prove that an additive track category with strong coproducts is equivalent to the category of pseudomodels for the algebraic theory of $\nil_2$ groups. This generalizes the classical statement that the category of models for the…

Gauging a finite Abelian normal subgroup $\Gamma$ of a nonanomalous 0-form symmetry $G$ of a theory in $(d+1)$D spacetime can yield an unconventional critical point if the original theory has a continuous transition where $\Gamma$ is…

This is the second part of a two part work in which we prove that for every finitely generated subgroup $\Gamma < \mathsf{Out}(F_n)$, either $\Gamma$ is virtually abelian or its second bounded cohomology $H^2_b(\Gamma;\mathbb{R})$ contains…

We define a notion of morphism for generalized affine buildings, also known as affine $\Lambda$-buildings, extending existing definitions and giving rise to a category of generalized affine buildings. For affine $\Lambda$-buildings equipped…

This article began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point…

We consider quotients of the Bruhat-Tits building associated to the projective linear groups of dimension $d>2$ over the function field $\mathbb F_q(t)$ by a non-uniform lattice $\Gamma$ which is a congruence subgroup in the non-uniform…

In the paper, it is proved that if a finite group $G$ possesses a $\pi$-Hall subgroup for a set $\pi$ of primes, then every normal subgroup $A$ of $G$ possesses a $\pi$-Hall subgroup $H$ such that ${G=AN_G(H)}$.

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places…