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Andrew W. Sale

Publications and source records attributed to Andrew W. Sale.

8 recordsLinked to original sources
arXiv2026

Conjugator length in finitely presented groups

The conjugator length function of a finitely generated group $G$ gives the minimal upper bound on the length of a conjugator for a pair of words that represent conjugate elements in $G$, as a function of the sum of the lengths of the words. Here, we seek to promote the systematic study of conjugator length functions by explaining their significance, by surveying what is known about them and by explaining fundamental techniques and examples.

math.GR↗
arXiv2025

Conjugacy in a family of free-by-cyclic groups

We analyse the geometry and complexity of the conjugacy problem in a family of free-by-cyclic groups $H_m=F_m\rtimes\mathbb{Z}$ where the defining free-group automorphism is positive and polynomially growing. We prove that the conjugator length function of $H_m$ is linear, and describe polynomial-time solutions to the conjugacy problem and conjugacy search problem in $H_m$.

math.GR↗
arXiv2020

Calculating the virtual cohomological dimension of the automorphism group of a RAAG

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.

math.GR↗
arXiv2014

Metric Behaviour of the Magnus Embedding

The classic Magnus embedding is a very effective tool in the study of abelian extensions of a finitely generated group $G$, allowing us to see the extension as a subgroup of a wreath product of a free abelian group with $G$. In particular, the embedding has proved to be useful when studying free solvable groups. An equivalent geometric definition of the Magnus embedding is constructed and it is used to show that it is 2-bi-Lipschitz, with respect to an obvious choice of generating sets. This is then applied to obtain a non-zero lower bound on $L_p$ compression exponents in free solvable groups.

math.GR↗
arXiv2013

Conjugacy Length in Group Extensions

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 behaviour 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 $\Z^d \rtimes \Z^k$ have at most exponential (if $k >1$) or linear (if $k=1$) conjugacy length functions.

math.GR↗
arXiv2013

The Geometry of the Conjugacy Problem in Wreath Products and Free Solvable Groups

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.

math.GR↗
arXiv2012

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

Let $G$ be a real semisimple Lie group with trivial centre and no compact factors. Given a conjugate pair of either real hyperbolic elements or unipotent elements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that $d_G(1,g) \leq L(d_G(1,u)+d_G(1,v))$, where $L$ is a positive constant which will depend on some property of $a$ and $b$. For the vast majority of such elements however, $L$ can be assumed to be a uniform constant.

math.GR↗