Andrew W. Sale

Palindromic width of wreath products, metabelian groups, and max-n solvable groups

Abstract A group has finite palindromic width if there exists n such that every element can be expressed as a product of n or fewer palindromic words. We show that if G has finite palindromic width with respect to some generating set, then so does G≀ℤ r

Conjugacy Length in Group Extensions

G \wr \mathbb {Z}^{r}

The geometry of the conjugacy problem in wreath products and free solvable groups

. We also give a new, self-contained proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.

Permute and conjugate: the conjugacy problem in relatively hyperbolic groups

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ℤd ⋊ ℤk have at most exponential (if k > 1) or linear (if k = 1) conjugacy length functions.

Vastness properties of automorphism groups of RAAGs

Abstract In this paper, 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 definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.

Metric behaviour of the Magnus embedding

Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words

Short Conjugators in Solvable Groups

u

The geometry of the conjugacy problem in lamplighter groups

and

Bounded Conjugators For Real Hyperbolic and Unipotent Elements in Semisimple Lie Groups

v

Metric Approximations of Wreath Products

, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy length function is bounded by the permutation conjugacy length function of the parabolic subgroups.