Related papers: Finite Free Entropy and Free Group Factors
We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.
We prove new separability results about free groups. Namely, if $H_1, \ldots , H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$, then there exists a homomorphism onto some alternating group $f:F…
Using Voiculsecu's free entropy, it is shown that the free group factors L(F_n) lack property C of Popa and that they lack finite multiplicity abelian subalgebras. Property C is an assymptotic commutivity property.
We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in $F_{\omega}$. We also prove that the set of primitive elements in finite rank free groups is not uniformly…
In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds…
We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant…
We study fundamental groups of non compact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.
Let $G$ be a discrete countable infinite group that does not have Kazhdan's property ~(T) and let $\kappa$ be a generating probability measure on $G$. Then for each $t>0$, there is a type $III_1$ ergodic free nonsingular $G$-action whose…
It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…
We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group is of…
We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.
In this paper, a notion of non-microstate bi-free entropy with respect to completely positive maps is constructed thereby extending the notions of non-microstate bi-free entropy and free entropy with respect to a completely positive map. By…
We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all…
We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not $\forall$-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated…
We study finite subgroups of outer automorphisms of free products. We give upper bounds for the orders of these finite subgroups as well as bounds for the orders of individual torsion outer automorphisms under some (necessary) conditions…
Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.
We show that any finitely generated group $F$ with infinitely many ends is not a group of fractions of any finitely generated proper subsemigroup $P$, that is $F$ cannot be expressed as a product $P P^{-1}$. In particular this solves a…
We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we…
We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary…
It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$,…