Related papers: A criterion for detecting trivial elements of Burn…

Given an infinite linear group with a finite set of generators, we show that the shortest word length of an element of infinite order has an upper bound that depends only on the number of generators and the degree. This provides a…

We show that Morse elements are generic in acylindrically hyperbolic groups. As an application, we observe that fully irreducible outer automorphisms are generic in the outer automorphism group of a finite-rank free group.

Let G be a torsion free hyperbolic group. We prove that the elementary theory of G is decidable and admits an effective quantifier elimination to boolean combination of AE-formulas. The existence of such quantifier elimination was…

In this article, for a polyadic group(G,f),derived from group G by automorphism G and element b, we give a necessary and sufficient condition in terms of the group, the automorphism G, and the element b, in order that the polyadic group…

We show that the theory of the free group -- and more generally the theory of any torsion-free hyperbolic group -- is $n$-ample for any $n\geq 1$. We give also an explicit description of the imaginary algebraic closure in free groups.

The topology of the Bowditch boundary of a relatively hyperbolic group pair gives information about relative splittings of the group. It is therefore interesting to ask if there is generic behavior of this boundary. The purpose of this…

We show that a group whose generalized torsion elements are torsion elements (which we call a $TR^{*}$-group) is torsion-by-$R^{*}$ group, an extension of torsion group by a group without generalized torsion elements. We also discuss a…

We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.

We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be…

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph…

We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups…

We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $\Gamma$…

We determine the average number of $3$-torsion elements in the ray class groups of fixed (integral) conductor $c$ of quadratic fields ordered by absolute discriminant, generalizing Davenport and Heilbronn's theorem on class groups. A…

We prove the decidability of the elementary theory of a free group.

We show that free-by-free groups satisfying a homological criterion, which we call excessive homology, are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply…

In this note we show that any homomorphism from a pure surface braid group to a torsion-free hyperbolic group either has a cyclic image or factors through a forgetful map. This extends and gives a new proof of an earlier result of the…

Let R be any ring (with 1), \Gamma a group and R\Gamma the corresponding group ring. Let H be a subgroup of \Gamma of finite index. Let M be an R\Gamma -module, whose restriction to RH is projective. Moore's conjecture: Assume for every…

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend…

We prove several results on the model theory of Artin groups, focusing on Artin groups which are ``far from right-angled Artin groups''. The first result is that if $\mathcal{C}$ is a class of Artin groups whose irreducible components are…

We obtain sufficient criteria for endomorphisms of torsion-free nilpotent groups of finite rank to be automorphisms, by considering the induced maps on the torsion-free abelianisation and the centre. Whilst these results are known in the…