Nonresidually Finite One-Relator Groups

Abstract

Abstract. The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form w1v'uvm which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.

Distortion and the automorphism group of a shift

“…On a related note, if m > 1 and n > m, then BS(m, n) is not residually finite [19]. Thus if X is a mixing shift of finite type, then BS(m, n) does not embed in Aut(X).…”

On the residual and profinite closures of commensurated subgroups

“…Corollary 25 (Meskin, [15]). The Baumslag-Solitar group BS(m, n) is residually finite if and only if the set {1, |m|, |n|} has at most 2 elements.…”

“…We will only prove one direction; namely, if BS(m, n) is residually finite, then |{1, |m|, |n|}| ≤ 2. See [15] for the converse.…”

Residual Finiteness of Quasi-Positive One-Relator Groups

“…The reason for scrutinizing this family of examples is that when n = 1 one obtains the presentations a, t | t −1 a p t = a q . The resulting finitely presented groups are residually finite if and only if either |p| = |q| or |p| 6 1 or |q| 6 1 [15], and are not Hopfian when gcd(p, q) = 1 [9]. Recall that a group G is Hopfian if every surjective endomorphism of G is injective.…”

“…Roughly speaking, Theorem 1.4 states that the one-relator group is residually finite if it satisfies C (1/n) and B is sufficiently longer than A. Theorem 1.4 fails definitively without the C (1/n) hypothesis. Indeed, one-relator groups of the form a, t | a m t = ta n are not residually finite unless either |n| = |m| or |n| 6 1 or |m| 6 1 [9,15].…”