Independence property and hyperbolic groups

We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.Comment: v3: 10 pages (11pt), a few typos corrected, minor rearrangements (e.g. Fact 2.3 and Lemma 2.5); v2: 8 pages (10pt), a false statement in the proof of Fact 2.4 is replaced with a true one; v1: 8 page

On certain equations of arbitrary length over torsion-free groups

Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold. Such equations appear for all lengths greater than or equal to eight and the results presented in this article can substantially simplify their solution.Comment: arXiv admin note: substantial text overlap with arXiv:1903.0650

Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length

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 vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.Comment: v4: 41 pages, 6 figures rescaled at 120%; references updated, typos corrected, other minor corrections. v3: minor changes to the title, text and figures. v2: 41 pages, 6 figures; correction: Ore's conjecture was proved in 2008; 2 references added. v1: 40 pages, 6 figure

On the torsion in a group F/[M,N]\bf F/[M,N] in the case of combinatorial asphericity of groups F/M\bf F/M and F/N\bf F/N

Let $F$ be a non-Abelian free group with basis $A$, $M$ and $N$ be the normal closures of sets $R_M$ and $R_N$ of words in the alphabet $A^{\pm 1}$. As is known, the group $F/[N, N]$ is torsion-free, but, in general, torsion in $F/[M, N]$ is possible. In the paper of Hartley and Kuz'min (1991), it was proved that if $R_M=\{v\}$, $R_N=\{w\}$ and words $v$ and $w$ are not a proper power in $F$, then $F/[M,N]$ is torsion-free. In the present paper a sufficient condition for the absence of torsion in $F/[M,N]$ is obtained, which allows to generalize the result of Hartley and Kuz'min to arbitrary words $v$ and $w$.Comment: arXiv admin note: text overlap with arXiv:1503.0619

Efficient Finite Groups Arising in the Study of Relative Asphericity

We study a class of two-generator two-relator groups, denoted Jn(m, k), that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups Fa,b,c with trivalent Cayley graphs that was introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to ?bicyclically presented groups? that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups