3-Manifold Topology, with Groups and Randomness

Abstract

In recent years there has been an increasing trend of interactions between different fields of mathematics, in particular, as a few major problems in low-dimensional topology gets solved this way, the attention has been shifting towards exploring the interactions between topology and various other fields. Note that such interaction can happen two-ways: one may apply a result or technique from another field to solve a problem in topology, or one could apply theorems or methods in topology to gain insights and solve problems in seemingly unrelated fields. In this thesis we will present results in both of the above mentioned directions. The first part of the thesis being an application of probability and Teichmuller theory to proving a purely topological theorem while the second part being applying classical topology to gain insights towards solving an open problem in group theory.

More precisely, the first result gave exact exponential growth rates for primi- tive elements in free groups with any number of generators by taking a topological viewpoint. The second result applies probability, among other things to prove the existence of hyperbolic homology 3-spheres with any given Heegaard genus g and Casson invariant n.

In light of the collaborative nature of the topic, I am extremely fortunate to have talked and worked with outstanding mathematicians from a variety of fields. The work in the first part is done in collaboration with Doron Puder while the second part with Alex Lubotzky and Joseph Maher.