Geometric group theory
My research is in geometric group theory. It's about using geometry (i.e. drawing pictures) to help us understand groups, which can otherwise be fairly dry algebraic objects (i.e. a bunch of letters on a piece of paper).
To expand on this in a more precise way, I am interested in algebraic and topological aspects of groups and explore many of these ideas using geometry. Common themes for my work involve rigidity problems (such as classifying maps between different families of groups or describing automorphism groups of complexes) and the study of classifying spaces of groups (such as constructing nice classifying spaces for groups or exploring the (co)homology of these classifying spaces). In particular I've been using classifying spaces to explore notions of duality in group theory.
Much of my work is on automorphism groups of free groups and, more generally, automorphism groups of right-angled Artin groups. Other areas of interest that are touched upon in the papers below are lower-central series and other central filtrations of groups, deformation spaces of trees, subspace arrangements, asymptotic dimension, duality groups, cohomological dimension, bounded cohomology, and combinatorial sheaves.
Preprints
The handlebody group is a virtual duality group
With Dan Petersen. arXiv.Problems on handlebody groups
With Naomi Andrew, Sebastian Hensel, and Sam Hughes. arXiv.Duality for Cohen–Macaulay Complexes through Combinatorial Sheaves
With Thomas Wasserman. arXiv.Published Articles
On the geometry of the free factor graph for \({\rm{Aut}}(F_N)\)
Groups, Geometry, and Dynamics, 2025.With Mladen Bestvina and Martin Bridson.
arXiv. Journal.
Cohen–Macaulay Complexes, Duality Groups, and the dualizing module of \( {\rm{Out}}(F_N)\)
IMRN, 2025.With Thomas Wasserman.
arXiv. Journal.
Commensurations of \({\rm{Aut}}(F_N)\) and its Torelli subgroup
Geometric and Functional Analysis, 2024.With Martin Bridson.
arXiv. Journal.
Direct products of free groups in \({\rm{Aut}}(F_N)\)
Geometric and Functional Analysis, 2024.With Martin Bridson.
arXiv. Journal.
Rigidity of the Torelli subgroup in \({\rm{Out}}(F_N)\).
Revista Matemática Iberoamericana, 2024.With Sebastian Hensel and Camille Horbez.
arXiv. Journal.
A note on virtual duality and automorphism groups of right-angled Artin groups
Glasgow Mathematical Journal, 2023.With an appendix by Benjamin Brück.
arXiv. Journal.
\( {\rm{Aut}}\)-invariant quasimorphisms on groups
Transactions of the AMS, 2023.With Francesco Fournier-Facio
arXiv. Journal.
Calculating the virtual cohomological dimension of the automorphism group of a RAAG.
Bulletin of the LMS, 2021.With Matthew Day and Andrew Sale.
arXiv. Journal.
The algorithm described in this paper was implemented in Python by Yutong Dai and is available here.
On the topological dimensions of some hyperbolic \({\rm{Out}}(F_N)\)-graphs
Pacific Journal of Mathematics, 2020.With Mladen Bestvina and Camille Horbez.
arXiv. Journal.
Commensurations of subgroups of Out.
Transactions of the AMS, 2020.With Camille Horbez.
arXiv. Journal.
Relative automorphism groups of right-angled Artin groups.
Journal of Topology, 2019.With Matthew Day.
arXiv. Journal.
Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups.
Groups, Geometry, and Dynamics, 2018.With Matthew Day.
arXiv. Journal.
The lower central series of a right-angled Artin group.
L’Enseignement Mathématique, 2015.arXiv. Journal.
Centralisers of Dehn twist automorphisms of free groups.
Proc. Cambridge Philos. Soc., 2015.With Moritz Rodenhausen.
arXiv. Journal.
Automorphisms of the graph of cyclic splittings of a free group.
Geometriae Dedicata, 2015.With Camille Horbez.
arXiv. Journal.
Folding free group automorphisms.
Q.J. Math, 2014.arXiv. Journal.
Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups.
Journal of the LMS, 2013.arXiv. Journal.
Actions of higher-rank lattices on free groups.
Compositio Mathematica, 2011.With Martin Bridson.
arXiv. Journal.