One-relator groups with torsion are conjugacy separable

We prove that one-relator groups with torsion are hereditarily conjugacy separable. Our argument is based on a combination of recent results of Dani Wise and the first author. As a corollary we obtain that any quasiconvex subgroup of a one-relator group with torsion is also conjugacy separable.Comment: 9 page

Freiheitss\"{a}tze for one-relator quotients of surface groups and of limit groups

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context.Comment: 17 page

The Outer Automorphism Groups of Two-Generator One-Relator Groups with Torsion

The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.Comment: 15 pages, final version. To appear in Proc. Amer. Math. So

Delzant's T-invariant, Kolmogorov complexity and one-relator groups

We prove that ``almost generically'' for a one-relator group Delzant's $T$-invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator groups and on the methods of the theory of Kolmogorov-Chaitin complexity. We also give a precise asymptotic estimate (when $k$ is fixed and $n$ goes to infinity) for the number $I_{k,n}$ of isomorphism classes of $k$-generator one-relator groups with a cyclically reduced defining relator of length $n$: $$I_{k,n}\sim \frac{(2k-1)^n}{nk!2^{k+1}}.$$ Here $f(n)\sim g(n)$ means that $\lim_{n\to\infty} f(n)/g(n)=1$.Comment: A revised version, to appear in Comment. Math. Hel