Research

The main theme of my research is the interaction between geometric structures on manifolds, geometric group theory, and discrete subgroups of Lie groups. Mostly, this means that I work with actions of groups with rich coarse geometry (such as hyperbolic and relatively hyperbolic groups) on homogeneous spaces (like symmetric spaces, or projective space and other flag manifolds). Important examples include convex cocompact and geometrically finite subgroups of SL(2, R) and SL(2, C), as well as higher-rank analogs, such as Anosov subgroups and their generalizations.

A quasifuchsian representation in H^3

The limit set of a convex cocompact representation in hyperbolic space (specifically, a quasifuchsian representation produced by bending). See an animated version.

Published and accepted articles

  1. (with M. Islam) Morse properties in convex projective geometry. Adv. Math., 479:110430, 2025.

    journal | download | arXiv

  2. (with S. Douba, B. Fléchelles, and F. Zhu) Cubulated hyperbolic groups admit Anosov representations, 2023. To appear in Geometry & Topology.

    download | arXiv

  3. (with A. Traaseth) Combination theorems for geometrically finite convergence groups. To appear in Algebraic and Geometric Topology.

    download | arXiv

  4. (with K. Mann and J.F. Manning) Stability of hyperbolic groups acting on their boundaries, 2022. To appear in Groups, Geometry, and Dynamics.

    journal | download | arXiv

  5. Dynamical properties of convex cocompact actions in projective space. J. Topol., 16(3):990-1047, 2023.

    journal | download | arXiv

Preprints

  1. Dehn filling in semisimple Lie groups. arXiv: 2502.17592, 2025.

    download | arXiv

  2. (with K. Tsouvalas) Singular value gap estimates for free products of semigroups. arXiv:2409.20330, 2024. Submitted

    download | arXiv

  3. (with A. Guilloux) Limits of limit sets in rank-one symmetric spaces. arXiv:2407.04301, 2024. Submitted.

    download | arXiv

  4. (with K. Mann and J.F. Manning) Topological stability of relatively hyperbolic groups acting on their boundaries. arXiv:2402.06144, 2024. Submitted.

    download | arXiv

  5. Examples of extended geometrically finite representations. arXiv:2311.18653, 2023.

    download | arXiv

  6. An extended definition of Anosov representation for relatively hyperbolic groups. arXiv:2205.07183, 2022. Submitted.

    download | arXiv

PhD thesis: Higher-rank generalizations of convex cocompact and geometrically finite dynamics.

A portion of the landscape of discrete subgroups of Lie groups

Part of the landscape of discrete subgroups of Lie groups with "intermediate" behavior between rank-one and higher rank. EGF representations are introduced in my paper "An extended definition of Anosov representation for relatively hyperbolic groups."

Talk slides and videos

Anosov representations of cubulated hyperbolic groups
slides

Dehn filling in semisimple Lie groups
slides

Combination theorems for geometrically finite convergence groups (NCNGT, June 2023)
video (part 1) | video (part 2)

Topological stability for (relatively) hyperbolic boundary actions (GTiNY, June 2023)
slides

Extended geometrically finite representations (STDC, March 2022)
slides

Extended convergence dynamics and relative Anosov representations (December 2021)
slides

Expansion/contraction dynamics for non-strictly convex projective manifolds (GTA Philadelphia, June 2021)
slides

Group actions on boundaries of convex divisible domains (November 2019)
slides

Gallery

Coxeter automaton

shortlex automatic structure for a 3,3,4 triangle group in H2
automaton for shortlex automatic structure on a 3,3,4 triangle group

A shortlex automatic structure for a (3,3,4) triangle group in the hyperbolic plane, drawn using my geometry_tools Python package. Each numbered vertex is a state in a finite state automaton, generated using the kbmag program.

This image was originally created for the postcard session of the 2021 Nearly Carbon Neutral Geometric Topology conference.

Figure-eight knot group automaton

inclusions for a single vertex of a relative automaton

Local structure of a finite-state "relative automaton" recognizing quasi-geodesics in the figure-eight knot group. This is an practical implementation of the construction used in arXiv:2205.07183 to prove relative stability properties of extended geometrically finite representations.

Pontryagin spheres

Pontryagin sphere in the boundary of 4-dimensional hyperbolic space

An approximation of the limit set of a convex cocompact reflection group in O(4,1), studied by Sami Douba, Gye-Seon Lee, Ludovic Marquis, and Lorenzo Ruffoni. The limit set of this group is a Pontryagin sphere, embedded equivariantly into the ideal boundary of 4-dimensional hyperbolic space (a 3-sphere).
Animated version (41MB file) | Interactive version (5MB file, less detail)
Deformation of a Bianchi group with Pontryagin sphere limit set
Deformation of a Bianchi group with Pontryagin sphere limit set
Deformation of a Bianchi group with Pontryagin sphere limit set
Limit sets of deformations of a Bianchi group in PU(3,1), projected to a 3-dimensional slice in the boundary of complex hyperbolic 3-space. The original group has 2-sphere limit set; it has arbitrarily small deformations which are not isomorphic to it, and have Pontryagin sphere limit set converging to this 2-sphere. See an animation.