# Past Junior Topology and Group Theory Seminar

We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.

A subgroup $H$ of a group $G$ is said to be engulfed if there is a

finite-index subgroup $K$ other than $G$ itself such that $H<K$, or

equivalently if $H$ is not dense in the profinite topology on $G$. In

this talk I will present a variety of methods for showing that a

subgroup of a discrete group is engulfed, and demonstrate how these

methods can be used to study finite-sheeted covering spaces of

topological spaces.