A generalization of weak
commutativity between two isomorphic groups

Lima, Bruno César Rodrigues; Sidki, Saïd

doi:10.1515/jgth-2016-0053

Cited by 2 publications

(1 citation statement)

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“…Related to X(G) one has the group [33], where D = [G, G] and L = ⟨{g −1 g g ∈ G}⟩. The circumstances under which E(G) is finitely presented are much more restrictive than for X(G); it is necessary but not sufficient that G be finitely presented -see [30].…”

Section: Introductionmentioning

confidence: 99%

Weak commutativity, virtually nilpotent groups, and Dehn functions

Bridson¹,

Kochloukova²

2022

Preprint

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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 = H 1 (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-fibered Kähler groups. The word problem is solvable in X(G) if and only if it is solvable in G. The Dehn function of X(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

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Section: Introductionmentioning

confidence: 99%