We will first introduce Summer UMS and then decide on the logistics for the next weeks. We will go over some potential textbooks we could cover and then pick one by vote. Every member will then have the opportunity to sign up to give a talk.

We will begin the talk with a general discussion of groups focusing on viewing them as the group of symmetries of some object. We will then talk about infinite groups, group homomorphisms as well as group presentations. Throughout this lecture, I will also mention some interesting examples such as the Coxeter groups, Braid groups, and Lamplighter groups.

In this talk, we will begin with a brief review of group actions and basic concepts of graphs. We will then introduce graph automorphisms, Cayley graphs, and, in particular, the Cayley graph of F_2. I will also mention how to introduce matrices on graphs and some relationships between geometric properties and algebraic properties of groups.

We'll define the fundamental group and see that the fundamental group of any connected finite graph is a free group. Then we'll show how, given any finitely-generated subgroup of a free group, we can construct a graph whose fundamental group is isomorphic to this subgroup; this construction will not only prove the famous result that every such subgroup is a free group but also answer specific algebraic questions about the subgroup using the properties of the graph.

We start by defining the automorphism group of a group, a way to describe the symmetries of a group itself; we address inner and outer automorphisms as well as a few finite cases. We then move to the specific case of the automorphism group of a free group, and in particular, the geometric description of the free group Fn as the rose on n petals. We use this picture to describe the *dynamics* of Aut(Fn); what the repeated application of a single automorphism does to a loop in the rose, by studying their lengths using tools from linear algebra.

What can be learned about a group by studying its action on a space? We encountered one response to this question in Office Hour 2: A group is torsion free if it has a free action by isometries on Euclidean space. The ping-pong lemma is another result in the same spirit but instead provides sufficient conditions for a group to be free. After I present the statement and proof of the ping-pong lemma, we take a short detour into the world of Möbius transformations and Schottky groups. Finally, I apply the ping-pong lemma to show that Schottky groups are free, a baby version of Maskit's 1967 "A characterization of Schottky groups."