Weak commutativity, virtually nilpotent groups, and Dehn functions

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 $\mathfrak{X}(G)$ 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-fibred Kähler groups. The word problem is soluble in $\mathfrak{X}(G)$ if and only if it is soluble 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.