On singular equations over torsion-free groups

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result has applications to equations over torsion-free groups: in particular a singular equation of syllable length at most [Formula: see text] over a torsion-free group has a solution in some overgroup.

To the question of existence of exact irreducible representations of soluble groups of finite rank over locally finite field

We find characteristic subgroup of soluble torsion-free group of finite rank, whose structure determines sufficient conditions of existence of exact irreducible representations of the group over locally finite field.

Levin Conjecture for Group Equations of Length 9

Levin conjecture states that every group equation is solvable over any torsion free group. The conjecture is shown to hold true for group equation of length seven using weight test and curvature distribution method. Recently, these methods are used to show that Levin conjecture is true for some group equations of length eight and nine modulo some exceptional cases. In this paper, we show that Levin conjecture holds true for a group equation of length nine modulo 2 exceptional cases. In addition, we present the list of cases that are still open for two more equations of length nine.

ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS

Let G be a nontrivial torsion-free group and $s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)$ be an equation over G containing no blocks of the form ${t^{- 1}}{g_i}{t^{ - 1}},{g_i} \in G$ . In this paper, we show that $s\left( t \right) = 1$ has a solution over G provided a single relation on coefficients of s(t) holds. We also generalize our results to equations containing higher powers of t. The later equations are also related to Kaplansky zero-divisor conjecture.

Cardinality of product sets in torsion-free groups and applications in group algebras

Let [Formula: see text] be a unique product group, i.e. for any two finite subsets [Formula: see text] of [Formula: see text], there exists [Formula: see text] which can be uniquely expressed as a product of an element of [Formula: see text] and an element of [Formula: see text]. We prove that if [Formula: see text] is a finite subset of [Formula: see text] containing the identity element such that [Formula: see text] is not abelian, then, for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Also, we prove that if [Formula: see text] is a finite subset containing the identity element of a torsion-free group [Formula: see text] such that [Formula: see text] and [Formula: see text] is not abelian, then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Moreover, if [Formula: see text] is not isomorphic to the Klein bottle group, i.e. the group with the presentation [Formula: see text], then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. The support of an element [Formula: see text] in a group algebra [Formula: see text] ([Formula: see text] is any field), denoted by [Formula: see text], is the set [Formula: see text]. By the latter result, we prove that if [Formula: see text] for some nonzero [Formula: see text] such that [Formula: see text], then [Formula: see text]. Also, we prove that if [Formula: see text] for some [Formula: see text] such that [Formula: see text], then [Formula: see text]. These results improve a part of results in Schweitzer [J. Group Theory 16(5) (2013) 667–693] and Dykema et al. [Exp. Math. 24 (2015) 326–338] to arbitrary fields, respectively.

On a Nonsingular Equation of Length 9 Over Torsion Free Groups

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

Topological rigidity for FJ by the infinite cyclic group

We call a group FJ if it satisfies the [Formula: see text]- and [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. We show that if [Formula: see text] is FJ, then the simple Borel conjecture (in dimensions [Formula: see text]) holds for every group of the form [Formula: see text]. If in addition [Formula: see text], which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions [Formula: see text]) holds for [Formula: see text]. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group [Formula: see text] satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text], then any semi-direct product [Formula: see text] also satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. Our result is indeed more general and implies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.