Carl-Fredrik Nyberg-Brodda - On the freeness problem for parabolic subgroups of the sepcial linear group

Abstract

I will report on some recent results on the freeness problem for parabolic subgroups of $SL(2;\mathbb{C})$.

Motivation

If we take two matrices, do they generate a free group in $S{L}_2(\mathbb{Q})$?

We are interested in $((1,0),(m,0))$, $((1,m),(1,0))$. For $m=2$ this is free.

Next up we look at $((1,0),(1,0))$ and $((1,r/p),(1,0))$. We call $r/p$ a relation number when the generator is not a free group. It describes how related the matrices are.

For $\frac{r}{p}\to 4$ the relations get longer and longer until at $4$ the group is free.

Making the problem easier?

We look at $SL(2,Z)$ but extend it by a prime $\frac{1}{p}$. What properties can we infer from the subgroup generated by $((1,0),(1,0))$ and $((1,r/p),(1,0))$. Is this subgroup free?