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.

Approximability of finite rank soluble groups by certain classes of finite groups

“…The simplest example of this descending HEN-extension is the Baumslang-Solitaire group G n = (a, b; b −1 ab = a n ), where n = 0. This group is finite approximable [9,10], and it is also almost F p -approximable if and only if p does not divide n [11].…”

Non-linear residually finite groups

“…Note only that all residually finite Baumslag-Solitar groups (i.e., HNN extensions of cyclic groups) [10] are linear [11].…”

Class-Preserving Automorphisms of Certain HNN Extensions of Baumslag-Solitar Groups

“…These groups have provided examples which mark boundaries between different classes of groups. For example, BS(λ, µ) is residually finite [19] if and only if |λ| = |µ| or |λ| = 1 or |µ| = 1. It is known that BS(λ, µ) is solvable if and only if |λ| = 1 or |µ| = 1.…”