2022/02/08 by Bridson, Martin R., Kochloukova, Dessislava H. · 1 citation
#20E22 (20F10 #20F18 #20F45) #20F65 #20J05 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2202.03796
The group \mathfrakX(G) is obtained from G∗ 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 \mathfrakX is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrakX(G). Subtle features of \mathfrakX are encoded in a normal abelian subgroup W<\mathfrakX(G) that is a module over ℤ Q, where Q= H1(G,ℤ). We establish a structural result for this module and illustrate its utility by proving that \mathfrakX preserves virtual nilpotence, the Engel condition, and growth type -- polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrakX(G) when G lies in a class that includes Thompson's group F and all non-fibered Kähler groups. The word problem is solvable in \mathfrakX(G) if and only if it is solvable in G. The Dehn function of \mathfrakX(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.