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Research Interests

My interests lie in the realm of geometric group theory. More precisely, I am interested in topological properties of infinite groups, including classifying spaces, finiteness properties, homological stability and Bieri-Neumann-Strebel-Renz invariants. A typical tool I use is discrete Morse theory, which turns difficult global problems into easier local ones. Some groups of interest include braid groups, the extended family of Thompson's groups, groups of automorphisms of free groups and algebraic and arithmetic groups. Some relevant topological spaces include arc complexes, poset geometries, CAT(0) cube complexes, Outer/Auter space and buildings.

My coauthors and collaborators:

Peter Abramenko
Eli Bashwinger
Jim Belk
Collin Bleak
Matt Brin
Kai-Uwe Bux
Martin Fluch
Andy Heald
Judy Holdener
Martin Kassabov
Lee Kennard
Yash Lodha
Marco Marschler
Francesco Matucci
Rob McEwen
Justin Tatch Moore
Mark Pearson
Lucas Sabalka
Eduard Schesler
Nick Scoville
Rachel Skipper
Rob Spahn
Marco Varisco
Stefan Witzel
I am supported by a Simons Collaboration Grant (award #635763, "Topological methods in geometric group theory," 2019-2024).

Using xkcd's word checker, here is a description of my research interests that uses only the 1000 most common words in the English language:

I study shapes and spaces, and try to understand groups that act on them. Some questions that I care about are whether a given group can act in a "nice" way on some "nice" space. The groups I like usually come with cool pictures and ways of viewing them. Some spaces I like are things like trees, and other flat shapes, and also "Outer space" and "buildings".