On the representation theory of finitely generated groups

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Groups are mathematical objects that encode symmetries. The integers are an important example of a group that encodes the symmetries of, say, a line of equally spaced cars on a straight highway. Linear groups, a main focus of this proposal, are those groups that, in a sense similar to the integer group example, encode symmetries of real life objects. Not all groups are linear, but when you can determine that a group is linear this opens up a whole new understanding of the group (in particular, you can apply all the tools from the subject of linear algebra). This proposal aims to understand the grey area that separates linear and nonlinear groups by forming numerical invariants and obstructions that stratify this gray area. While this proposal will give insight in the theory of groups, it also has applications to computer science and interfaces algebraic geometry and geometric group theory. Further, the proposed work will impact undergraduate and graduate students at the City College of New York (CCNY) through mentoring on topics related to this proposal. This proposal's primary focus is on the complexity of constructing homomorphisms from a fixed group to a group of invertible matrices with complex entries. This focus builds on previous research, where the PI explored the interaction of Gromov's word growth and Lubotzky-Segal's subgroup growth by developing the program of quantifying residual finiteness. This program gives a systematic approach for studying the complexity of constructing homomorphisms from a fixed group to any finite group (which is also linear). Further, this program paves way for the discovery of new characterizations of fundamental properties of finitely generated groups including nilpotency, linearity, arithmeticity, and even finite presentability. While continuing the pursuit of quantifying residual finiteness, the PI will augment his current research program by detailing the behavior of homomorphisms from a fixed group into a semisimple linear algebraic group (i.e., a special linear group). In particular, the PI seeks to determine when Borel's theorem on free groups generalizes to other groups. This research direction would influence the understanding of the behavior of arithmetic groups and word maps. Arithmetic groups are central in geometry because they supply explicit examples of manifolds. The study of word maps finds its motivation from the proof of the Hausdorff-Banach-Tarski paradox and is intertwined with the theory of expander graphs (a subject of great interest in theoretical computer science). While the projects proposed are grounded in topology, they feature methods from algebraic geometry, group theory, and number theory.