Geometric Group Theory, Random Graphs, and Isoperimetry

Research Foundation Of The City University Of New York (Lehman), Bronx NY

Investigators

Abstract

Non-positive curvature is the study of rapid growth, such as that occurring in population explosions, the shape of coral reefs, and connections in real-world networks (e.g., neural-anatomy of the brain and the structure of the internet). This project involves several distinct areas, each with connections to Geometric Group Theory, in which the PI will develop and apply new tools and techniques for understanding spaces with various aspects of non-positive curvature. One aspect of the project will involve new techniques for studying a wide range of spaces that are "hierarchically hyperbolic"; these spaces were introduced by the PI together with M. Hagen and A. Sisto and generalize Gromov's highly successfully notion of a "delta-hyperbolic space" to a much broader context. The PI intends to further develop this framework with a number of applications, including one involving a new method for building walls. Another part of the project will study "random graphs" through techniques which the PI and his collaborators have been developing. Random graphs have both interesting theoretical and applied interest; one goal is to resolve a particular conjecture in percolation theory. This project studies Geometric Group Theory and its interactions with surrounding areas. The groups studied are interesting both by themselves and in establishing a paradigm for more general classes of groups (e.g., CAT(0) groups, outer automorphism groups, relatively hyperbolic groups). The three main aspects of this project are to: continue developing Erdos-Renyi-style "threshold theorems'' in random graphs (joint with V. Falgas-Ravry and T. Susse); study higher-isoperimetric functions in groups (joint with C. Drutu); and, study algebraic and geometric properties of hierarchically hyperbolic groups and spaces (joint with M. Hagen and A. Sisto). In the project the PI will work to establish models for studying broad classes of groups and spaces while also strengthening the interplay between geometric group theory and surrounding fields including combinatorics and geometric measure theory.