Abstract: A triangle group is a group of the form $G = \langle x,y| x^{p_1}, y^{p_2}, (xy)^{p_3}\rangle$. If $1/p_1+1/p_2+1/p_3 < 1$, then $G$ admits a representation into $PSL(2,\mathbb{R})$. These groups are well-studied and provide a wealth of interesting examples, such as if $p_1=2$, $p_2=3$ and $p_3 = \infty$, we get the modular group $PSL(2,\mathbb{Z})$. The first talk will be a discussion of hyperbolic triangle groups and how to get nice $PSL(2,\mathbb{R})$ representations. While performing the construction, we will pay attention to the arithmetic data we can associate with these simple representations and end on a construction of congruence quotients for these groups. The second talk will briefly discuss a problem in 3-manifold topology in service of restating and then resolving it entirely in terms of how prime ideals split in a restricted class of number fields. This is joint work with Kate Petersen. |