Quasiperiodic and mixed commutator factorizations in free products of groups

Abstract

It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation [x1, y1] . . . [x k , y k ] = z n , where n 2k, in the free product F of groups without nontrivial elements of order n implies that z is conjugate to an element of a free factor of F . If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct. m j=1 n j . Culler [8] discovered that, in the free group F (a, b) with free generators a, b, the cube [a, b] 3 of the commutator [a, b] := a −1 b −1 ab is a product of two commutators, [a, b] 3 = [a −1 ba, a −2 bab −1 ][bab −1 , b 2 ].Moreover, [a, b] n is a product of k commutators whenever n 2k − 1, see [8].

Spectral gap of scl in free products

“…Finally, Corollary 2.2 and Theorem 3.1 together imply the following result about commutator length similar to the results obtained by Ivanov-Klyachko [12]: Corollary 3.7. Let G = A * B and g = a 1 b 1 · · · a L b L with a i ∈ A\{id}, b i ∈ B\{id} and L ≥ 1 such that g ∈ [G, G].…”

“…In fact we give two logically independent proofs of Corollary B. Ivanov-Klyachko [12] recently independently obtained Theorem A together with other estimates related to Theorem 3.1 in terms of commutator length. Their argument uses a different language (diagrams) but the idea behind is similar, especially in the case of free groups.…”

Gaps in scl for Amalgamated Free Products and RAAGs

“…• Sometimes, one may control scl on certain generic group elements. If G = G 1 G 2 is the free product of two torsion-free groups G 1 and G 2 and g ∈ G does not conjugate into one of the factors, then scl(g) ≥ 1/2; see [Che18] and [IK17]. Similarly, if G = A C B and g ∈ G does not conjugate into one of the factors and such that CgC does not contain a copy of any conjugate of g −1 then scl(g) ≥ 1/12.…”

Powers with minimal commutator length in free products of groups

“…Moreover, they proved a similar assertion for free products of locally indicable groups: if g is an element of a free product of locally indicable groups such that g is not conjugate to elements of the free factors, then cl(g n ) ≥ [n / 2] + 1. This assertion turned out to be true in a free product of arbitrary torsion-free groups, it was independently discovered by Ivanov and Klyachko [9] and Chen [2].…”