Spectral gap of scl in free products

Commutator length of powers in free products of groups

“…This was shown in [2,19] (independently). Moreover, both papers mentioned that the arguments remain valid if the torsion-free condition is replaced with a small-torsion-free condition.…”

“…For N (G) = 2, the assertion holds, because the group G in this case contains an infinite dihedral subgroup, whose commutator subgroup coincides with the set of commutators (and trivially intersects conjugates of free factors). For N (G) = ∞, the argument below is essentially valid, but we leave it to readers, because the assertion of the theorem in this case follows immediately from the results of [2,19] mentioned in the introduction. Thus, we assume that N (G) is odd.…”

“…It is well known that, in free groups, non-identity commutators are not proper powers [29]. A product of two commutators in a free group can surely be the square of a non-identity element, and can even be a cube, as Culler noticed [9] 2 ]. This equality holds in the free group F (a, b) and, therefore, for any elements a and b of any group.…”

“…Moreover, both papers mentioned that the arguments remain valid if the torsion-free condition is replaced with a small-torsion-free condition. However, arguments in [2,19] are different: -Chen's proof is based on Calegari's approach [1],…”

“…-while [19] is based on the car-crash lemma [20]; this is why, the results of [2,19] for groups with torsion are different (and even incomparable -neither one is stronger than the other):…”