Topology Seminar

For a (finitely generated) group $G$ and a given (finite) set of generators $\mathcal{S}$, one can define the growth function $\sigma_{\mathcal{S}}(n)$ to be the function that spits out the number of group elements of length $n$. From this we can take a generating function where the coefficient of $x^n$ is $\sigma(n)$. In doing so, one can ask if the generating function is a rational function, in which case we say the group has rational growth in that generating set, which implies several nice properties about the language of geodesics.

One class of groups in which growth is studied is the class of Baumslag-Solitar groups $BS(p,q)$. I will particularly focus on $BS(1,3)$, which is a solvable group that has rational growth in standard generators. In this talk I will define all of these things I said above, as well as give a class of HNN extensions generalizing Baumslag-Solitar that we call the higher Baumslag-Solitar groups. The main result will be the rationality of the growth of a particular solvable class of higher Baumslag-Solitar groups generalizing $BS(1,3)$. Time permitting, I will discuss some future directions.
This work is joint with Michael Shapiro. Speaker(s): Ayla Sanchez (Wheaton College)