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The purpose of this paper is two fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor…
We characterize which groups splitting as finite graphs of free groups with cyclic edge groups are residually finite. Such a group $G$ is residually finite if and only if all its Baumslag-Solitar subgroups are residually finite. From a…
We prove Roth type theorems in finite groups. Our main tool is the Triangle Removal Lemma of Ruzsa and Szemer\'edi.
It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of…
We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann.…
We construct examples of finitely generated infinite simple groups of homeomorphisms of the real line. Equivalently, these are examples of finitely generated simple left (or right) orderable groups. This answers a well known open question…
In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The…
We prove that a finitely generated group $G$ is virtually free if and only if there exists a generating set for $G$ and $k > 0$ such that all $k$-locally geodesic words with respect to that generating set are geodesic.
We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy…
We prove that there exist finitely generated, stably finite algebras which are non linear sofic. This was left open by Arzhantseva and P\u{a}unescu in 2017.
A theorem of Myasnikov and Roman'kov says that any verbally closed subgroup of a finitely generated free group is a retract. We prove that all free (and many virtually free) verbally closed subgroups are retracts in any finitely generated…
We fill details in the proof of Lemma 13 in Bull.London.Math.Soc 40 (2008) (that is Lemma 3.2 in arXiv:0712.4244v1).
We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It…
We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_{\Gamma} : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_{\Gamma}$ is bounded iff…
In his $1994$ survey, Kleinert defined formally and formulated the problem to obtain unit theorems for unit groups of orders in a semisimple algebra $A$. If $A$ is a group algebra $FG$, it boils down to classifying all finite groups $G$…
Let $G$ be a virtually special group. Then the residual finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $\text{SL}_k(\mathbb{Z})$, for $k…
We give a technical result that implies a straightforward necessary and sufficient conditions for a graph of groups with virtually cyclic edge groups to be one ended. For arbitrary graphs of groups, we show that if their fundamental group…
We generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten's notion of…
In a series of papers starting in [Sel01] and culminating in [Sel07], Z. Sela proved that free groups, and more generally torsion-free hyperbolic groups, have a stable first-order theory. The question of the stability of the free product of…
We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…