Dissertations, Theses, and Capstone Projects

Date of Degree

9-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Olga Kharlampovich

Committee Members

Alina Vdovina

Vladimir Shpilrain

Subject Categories

Physical Sciences and Mathematics

Keywords

Group Theory, Random Groups

Abstract

In this dissertation, we delve into the subjects that lie in the intersection of random group theory and first-order theory, or as traditionally referred to in the field, the first-order theory of free groups (Sela (2006) and Kharlampovich and Myasnikov (2006), among others). In particular, we establish the following two findings. First, we demonstrate that for fixed k > 2, among finitely generated groups, those without finite index subgroups of index k are prevalent in the density model (as discussed in Chapter 3). Second, we show that a first-order universal sentence in the language of groups is almost surely true in the few-relator model if and only if it is true in a non-abelian free group (Chapter 4). The methodology used to prove the latter result draws significant inspiration from Ollivier’s proof that random groups in the density model of density less than 1/2 (Ollivier (2005)) are almost surely hyperbolic, as well as the techniques employed in Kharlampovich and Sklinos (2022) to establish a similar result about universal sentences for groups in the density model. An overview of Ollivier’s proof, certain group properties in random groups and the study of their behavior within different theoretical frameworks.

Chapter 1 This introductory chapter is devoted to free groups, random groups, hyperbolicity, and small cancellation properties of groups. It begins by defining free groups constructed from finite sets of generators, and highlights a few of their properties that are used later. The chapter then introduces the concept of random groups, outlining three distinct models for their construction: the few-relator model, the few-relator model with various lengths, and the density model. These models serve as the basis for studying random groups and their properties.

The discussion also turns to hyperbolicity, which is examined from both a word-based perspective, emphasizing Van Kampen diagrams, and a geometric viewpoint involving geodesic triangles in Cayley graphs. The chapter concludes with a discussion about small cancellation properties, which shed light on the relationships between relators in group presentations.

Chapter 2 In this chapter, we follow the proof outlined by Yann Ollivier for the key theorem, initially put forth in Mikhael Gromov’s work, that reveals the hyperbolic nature of random groups at density less than 1/2. The connection between the probabilistic construction of groups and their geometric properties, which is emphasized in this proof, will serve as the foundation for our work in Chapter 4 when we are faced with the need to establish the hyperbolicity of random groups. By examining and employing abstract van Kampen diagrams and isoperimetric inequalities, this proof aims to demonstrate that random groups, under specified density conditions, exhibit hyperbolic characteristics with overwhelming probability. The methodology brought about by this proof bridges the gap between theoretical probability and geometric intuition, offering new insights into the structure of random groups.

Chapter 3 This chapter explores random groups at a given density, aiming to establish the absence, with overwhelming probability, of finite index subgroups of specified order. We make use of mathematical ideas, such as non-backtracking random walks, to gain insights into how random words of length ℓ are distributed in finite groups with a fixed order. Additionally, the density model setting provides us with the ability to set a bound on the number of relators based on their length, which is crucial for our probability computations. Conversely, similar computations are not feasible for the random groups within the few-relator model due to the absence of such a bound. Our results offer insight into the probability of finding a subgroup of fixed index in random groups with density less then 1.

Chapter 4 The primary objective of this chapter is to demonstrate the following property: every solution of a given system of equations in a random group in the few-relator model comes from a solution in the free group. This proof highlights the connection between random groups in the few-relator model and non-abelian free groups, particularly concerning universal sentences. Leveraging the small-cancellation property in random groups, our work involves the use of Van-Kampen diagrams in small cancellation groups, as well as an indepth analysis of geodesics within these groups. Through this work, we hope to contribute to a deeper understanding of the intricate relationships between algebraic structures and geometric properties of random groups.

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