My research interests lie in low-dimensional geometry and topology. More precisely, I study hyperbolic geometry (mostly complex), reflection groups, and lattices in rank 1 semisimple Lie groups.
A tessellation is a way of filling space with non-overlapping tiles in a pattern that repeats infinitely often. A lattice is the symmetry group of a tessellation. For example, the Euclidean plane (or 3-space, or higher) can be tessellated by squares (or cubes, or hypercubes), and the corresponding lattice is a product of infinite cyclic groups. The full understanding of all crystallographic structures in 3 dimensions is crucial in Chemistry.
Analogous tessellations in hyperbolic spaces are much more abundant and much less understood. Hyperbolic spaces are spaces with negative curvature, which means loosely that non-intersecting lines diverge from each other in both directions. These spaces appear naturally in special relativity, as Lorentz and Minkowski space-times.
The picture on the right shows a tessellation of hyperbolic 3-space by regular right-angled dodecahedra.
My research is currently supported by the National Science Foundation (Grant DMS 1708463 ), Cartel Coffee Lab and Press Coffee.
Seminar: Our Geometry seminar is currently meeting Fridays 12-1pm in WXLR A107.
Preprints and work in progress:
- (with J. Wells) Hybrid lattices and thin subgroups of Picard modular groups. Preprint (2018).
- (with A. Mark) Presentations for cusped arithmetic hyperbolic lattices. Preprint (2017).
- (with M. Deraux and J. Parker) New non-arithmetic complex hyperbolic lattices II. Preprint (2016).
- (with P. Will) Involution and commutator length for complex hyperbolic isometries. Michigan Math. J. 66 (2017), 699-744. Preprint.
- (with P. Will) Real reflections, commutators and cross-ratios in complex hyperbolic space. Groups Geom. Dyn. 11 (2017), 311-352. Preprint.
- (with M. Deraux and J. Parker) New non-arithmetic complex hyperbolic lattices. Invent. Math. 203 (2016), 681-771. Preprint, online version.
- A simple method to compute volumes of even-dimensional Coxeter polyhedra, in In the Tradition of Ahlfors-Bers, VI, Contemporary Mathematics, vol. 590, Amer. Math. Soc., Providence RI, 2013, pp. 167-175. Preprint. Erratum.
- Non-discrete hybrids in SU(2,1). Geom. Dedicata 157 (2012), 259-268. Preprint.
- (with M. Deraux and J. Parker) Census of the complex hyperbolic sporadic triangle groups. Experiment. Math. 20 (2011), 467-486. Preprint.
complex hyperbolic triangle groups III: arithmeticity and commensurability . Pacific J. Maths. 245 (2010), 359-372. Preprint.
- (with J. Parker) Unfaithful
complex hyperbolic triangle groups II: Higher order
reflections. Pacific J. Maths. 239 (2009), 357-389. Preprint.
- Applications moment, polygones de configuration et groupes discrets de réflexions complexes dans PU(2,1). Séminaire de théorie spectrale et géométrie, Grenoble. TSG 24 (2007), 45-60. Preprint.
- Elliptic triangle groups
in PU(2,1), Lagrangian triples and momentum
maps. Topology 46 (2007), 155-183. Preprint.
- (with M. Deraux and E. Falbel) New constructions of fundamental polyhedra in complex hyperbolic space. Acta Math. 194 (2005), 155-201. Preprint.
- (with E. Falbel) Fundamental domains for finite subgroups in U(2) and configurations of Lagrangians. Geom. Dedicata 109 (2004), 221-238. Preprint.
- Configurations of Lagrangians, fundamental domains and discrete subgroups of PU(2,1), Ph.D. Thesis, Université Paris 6, 2005 (advisor: Elisha Falbel), pdf.
- Thesis abstract, pdf.