Related papers: On singular equations over torsion-free groups
We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups.
A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length…
A classical theorem of Roth states that the maximum size of a solution-free set of a homogeneous linear equation $\mathcal{L}$ in $\mathbb{F}_p$ is $o(p)$ if and only if the sum of the coefficients of $\mathcal{L}$ is $0$. In this paper, we…
Torsion theories are a pinnacle in the theory of abelian categories. They are a generalization of torsion abelian groups and in this generalization one of the most studied is that whose torsionfree class consists of nonsingular modules. To…
We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.
In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of…
It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an…
We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and…
We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to…
The main result includes as special cases on the one hand, the Gerstenhaber--Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii--Howie--Short theorem (1980--1984) generalising…
This paper classifies the pairs of nonzero integers $(m,n)$ for which the locally compact group of combinatorial automorphisms, Aut$(X_{m,n})$, contains incommensurable torsion-free lattices, where $X_{m,n}$ is the combinatorial model for…
We continue our work on the model theory of free lattices, solving two of the main open problems from our first paper on the subject. Our main result is that the universal (existential) theory of infinite free lattices is decidable. Our…
New upper bounds on the size of the torsion group of a $\mathbb{Q}$-acyclic simplicial complex are introduced which depend only on the vertex degree sequence of the complex and its dimension.
In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in…
We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of…
We investigate the properties of word lengths of elements from a three-reflection symmetric generating set of the dihedral group $D_n$. Specifically, we provide the upper bound $\lambda_1(D_n,S) \leq \lfloor\frac{n}{2}\rfloor + 1$ for a…
We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche-Wendt groups.
We determine which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero. This…
We show the existence of finitely presented torsion-free groups with decidable word problem that cannot be embedded in any finitely generated group with decidable conjugacy problem. This answers a well-known question of Collins from the…
We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov and give an estimate of their asymptotic dimension in terms of the relator length.