** "Geometry and Analysis on Groups" Research Seminar **

Talk 1: "Free Subgroups of Small Cancellation Groups"

Small cancellation is a property for group presentations that has several geometric and algebraic applications. We will begin by defining various "versions" of small cancellation theory. We will try to understand how one can use it to conclude that certain subgroups of small cancellation groups are free. We will also talk about some (very) interesting open questions. There will be lots of examples!

Talk 2: "Nielsen Equivalence Classes in a Class of Random Groups"

Nielsen introduced the idea of Nielsen Transformations to study subgroups of free groups. We will define the notion of Nielsen Equivalence as an equivalence relation defined on the set \(G^n\) of ordered \(n\)-tuples for any group \(G\). In general, it is extremely hard to distinguish between generating \(n\)-tuples for a group \(G\). We will state a result of Kapovich and Weidmann relating to Nielsen equivalence classes of generating \(n\)-tuples in random groups, and then generalize it "in a certain sense". Many of the proof ideas will emulate those introduced in Talk 1.