Weak commutativity, virtually nilpotent groups, and Dehn functions

TL;DR

This paper studies a group construction that preserves key algebraic and geometric properties, analyzes its word problem complexity, and establishes bounds on its Dehn function, with applications to various classes of groups.

Contribution

It introduces new properties of the functor x, proves preservation of virtual nilpotence and growth types, and provides isoperimetric bounds for the resulting groups.

Findings

x preserves virtual nilpotence and growth type.

The word problem solvability is equivalent in G and x(G).

Dehn function is at least cubic if G maps onto a non-abelian free group.

Abstract

The group $\mathfrak{X}(G)$ is obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. We make significant additions to the list of properties that the functor $\mathfrak{X}$ is known to preserve. We also investigate the geometry and complexity of the word problem for $\mathfrak{X}(G)$. Subtle features of $\mathfrak{X}$ are encoded in a normal abelian subgroup $W<\mathfrak{X}(G)$ that is a module over $\mathbb{Z} Q$, where $Q= H_1(G,\mathbb{Z})$. We establish a structural result for this module and illustrate its utility by proving that $\mathfrak{X}$ preserves virtual nilpotence, the Engel condition, and growth type -- polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for $\mathfrak{X}(G)$ when $G$ lies in a class that includes Thompson's group $F$ and all non-fibered K\"ahler groups. The word problem is solvable in $\mathfrak{X}(G)$ if and only if it is solvable in $G$. The Dehn function of $\mathfrak{X}(G)$ is bounded below by a cubic polynomial if $G$ maps onto a non-abelian free group.