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- Title
Weak commutativity, virtually nilpotent groups, and Dehn functions.
- Authors
Bridson, Martin R.; Kochloukova, Dessislava H.
- Abstract
The group X(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 X is known to preserve. We also investigate the geometry and complexity of the word problem for X(G). Subtle features of X(G) are encoded in a normal abelian subgroup W<X(G) that is a module over ZQ, where Q=H1(G,Z). We establish a structural result for this module and illustrate its utility by proving that X preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for 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 X(G) if and only if it is soluble in G. The Dehn function of X(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.
- Subjects
NONABELIAN groups; ISOPERIMETRIC inequalities; NILPOTENT groups; FREE groups; POLYNOMIALS
- Publication
Commentarii Mathematici Helvetici, 2024, Vol 99, Issue 1, p1
- ISSN
0010-2571
- Publication type
Article
- DOI
10.4171/CMH/563