Spread and infinite groups

Abstract: My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are many natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, which Scott Harper recently talked about at this seminar (and mentioned several interesting questions that he and Casey Donoven posed for infinite groups in arxiv:1907.05498. A group $G$ has spread $f$ if for every $g_1,\ldots,g_k$ we can find an $h$ in $G$ such that $ G = < g_i,h > $. For any group we can say that if it has a proper quotient that is non-cyclic, then it cannot have positive spread. 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: it is also a sufficient one. But 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. So what if we restrict ourselves to 2-generated groups? In this talk we’ll see the answer to this question. The arguments will be concrete -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- and accessible to a general audience.

group theoryrings and algebras

Audience: researchers in the topic

Al@Bicocca take-away