Thin groups as monodromy groups, Part I - Jordan Ellenberg (University of Wisconsin-Madison)

This lecture explores the occurrence and significance of thin groups as monodromy groups arising naturally in algebraic and arithmetic geometry. It focuses on explaining how monodromy groups form from fibrations, how these groups act on topological and algebraic structures, and the implications for understanding the size and nature of such groups. The talk is aimed at mathematicians interested in the interaction between topology, algebraic geometry, and group theory, particularly those curious about thin groups from a consumer perspective rather than a producer of results.

Overview of Monodromy and its Topological Origins

Monodromy captures the behavior of traveling around loops in a base space and examining how fibers of a fibration transform. The basic example discussed is a finite covering space \(X \to B\) over a base \(B\), often a sphere with punctures. Over each point of \(B\), the fiber \(X_b\) is a finite set of points.

• Topological description: Locally around a contractible neighborhood on \(B\), the preimage looks like disjoint copies of that neighborhood—hence a finite cover.
• Monodromy action: Tracing a loop in \(B\) permutes the points in the fiber above a basepoint \(b\), defining a homomorphism from the fundamental group \(\pi_1(B, b)\) into the symmetric group on the fiber points.
• There is a fundamental correspondence: finite covers correspond to representations of the fundamental group into finite permutation groups.

Transitivity of the monodromy action corresponds to connectedness of the covering space, and connected covers produce a transitive image in the symmetric group, leading to well-defined stabilizer subgroups that connect with classical Galois correspondences.