Randomizing and Describing Groups

In this work we study some topics in logic and group theory, and naturally, their intersection. One important theme in this work is the notion of random groups, which is informally the study of “what properties do ‘most’ groups satisfy?” In Chapter 2, we are interested in the theory of a random group, i.e., the properties we are interested here are first-order properties. Knight conjectured that the limit of the theories of random groups should converge to the theory of the free groups. In this chapter, we establish some partial results in the one-quantifier case. In Chapter 3, instead of looking at a random group, we focus our attention on only the class of nilpotent groups, and look at a random nilpotent group. We analyze the distribution of ranks of random nilpotent groups, and establish some threshold result on when a random nilpotent group will be finite and trivial. In Chapter 4, we turn our attention to regular (non-random) groups, and we study the complexities of Scott sentences and the index sets of groups. We show that many groups that are “tame” in the sense of group theory have low-complexity Scott sentences and index sets.