Spread and infinite groups

Abstract: My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups (in arxiv.org/abs/1907.05498). A group $G$ has spread $k$ if for every $g_1, \dots, g_k \in G$ we can find an $h \in G$ such that $\langle g_i, h \rangle = G$. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread 0. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven’s first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we’ll see the answer to this question. The arguments will be concrete (*) and accessible to a general audience.

(*) at the risk of ruining the punchline, we will find a 2-generated group that has every proper quotient cyclic but that has spread zero.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle