Magnus pairs in, and free conjugacy separability of, limit groups

There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable

Bianchi groups are conjugacy separable

We prove that non-uniform arithmetic lattices of $SL_2(\mathbb{C})$ and in particular the Bianchi groups are conjugacy separable. The proof based on recent deep results of Agol, Long, Reid and Minasyan.Comment: to appear in J. Pure and Applied Algebr

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.Comment: 11 pages, 1 figure. Section 3 is replaced by Theorem 1.3. More detailed arguments. Added new references. To appear in Journal of Pure and Applied Algebr

Finite Affine Groups: Cycle Indices, Hall-Littlewood Polynomials, and Probabilistic Algorithms

The asymptotic study of the conjugacy classes of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given--three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group.Comment: Revised version, to appear in J. Algebra. A few typos are fixed; no substantive change