Related papers: All groups are surjunctive

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular,…

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every…

We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic…

We prove that for every finitely generated subgroup of a virtually connected Lie group which admits a finite dimensional model for the classifying space for proper actions the assembly map in algebraic K-theory is split injective. We also…

We show that the Strong Novikov Conjecture for the maximal C*-algebra C*(G) of a discrete group G is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show…

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of…

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated…

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group $G$ the subgroup $\gamma_{k}(G)$ is…

We establish generalizations of the well-known surjunctivity theorem of Gromov and Weiss as well as the dual-surjunctivity theorem of Capobianco, Kari and Taati for cellular automata (CA) to local perturbations of CA over sofic group…

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group G in this category is a cover of a definable group. We…

We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model…

We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group actions and discuss several applications. We prove in particular that for any group $\Gamma$ and any field $\K$, the space of…

Let $G$ be the linear algebraic group $SL_3$ over a field $k$ of characteristic two. Let $A$ be a finitely generated commutative $k$-algebra on which $G$ acts rationally by $k$-algebra automorphisms. We show that the full cohomology ring…

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq…

Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is…

We prove that the isomorphism problem for finitely generated fully residually free groups is decidable. We also show that each finitely generated fully residually free group G has a decomposition that is invariant under automorphisms of G,…

In this paper we prove that every $2$-generator finite $p$-group $G$ has a non-inner automorphism of order $p$ leaving $G^p\gamma_4(G)$ elementwise fixed ($p\ge 5$). Moreover, we prove a $2$-generator finite $3$-group satisfying…