Presentations for cusped arithmetic hyperbolic lattices

We present a general method to compute a presentation for any cusped hyperbolic lattice $\Gamma$, applying a classical result of Macbeath to a suitable $\Gamma$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ for $d=1,3,7$ and the quaternionic lattice ${\rm PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$.Comment: 25 pages, 4 figures, 6 tables. Version 2: added relations to presentations for d=3 and 7. Version 3: corrected a typo in the presentation for the Hurwitz modular group. Version 4: Made changes in response to a referee report. Found new points, resulting in a few more relations being added to the presentation of the Hurwitz modular grou

Involution and commutator length for complex hyperbolic isometries

We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \geqslant 3$ PU($n$,1) has involution length at most 8.Comment: 32 pages, 22 figure

Non-discrete hybrids in SU(2, 1)

We show that a natural hybridation construction of lattices in SU(n, 1) to produce (non- arithmetic) lattices in SU(n + 1, 1) fails when n = 1 for most triangle groups in SU(1, 1)

Non-discrete hybrids in SU(2, 1)

We show that a natural hybridation construction of lattices in SU(n, 1) to produce (non-arithmetic) lattices in SU(n+1, 1) fails when n=1 for most triangle groups in SU(1, 1

Real reflections, commutators and cross-ratios in complex hyperbolic space

26 pagesInternational audienceWe provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8

Configurations de lagrangiens, domaines fondamentaux et sous-groupes discrets de PU(2,1).

The object of this thesis is to investigate discrete subgroups of$PU(2,1)$, the group of holomorphic isometries of complex hyperbolic space of (complex) dimension 2. We are mostly concerned with those groups generated by elliptic motions, i.e. motions fixing a point of this space.The two guiding principles of this work are on one hand the use ofLagrangian subspaces (or real planes) and their associated reflections (antiholomorphic involutions), and on the other hand the study and understanding of the examples of lattices in $PU(2,1)$ which Mostow constructed in 1980.L'objet de cette thèse est l'étude de sous-groupes discrets de$PU(2,1)$, groupe des isométries holomorphes de l'espace hyperbolique complexe de dimension (complexe) 2. On s'intéresse en particulier aux groupes engendrés par des transformations elliptiques, i.e. ayant un point fixe dans cet espace. Les deux fils conducteurs de ce travail sont d'une part l'utilisation des sous-espaces lagrangiens (ou plans réels) ainsi que des réflexions associées (des involutions antiholomorphes), et de l'autrel'étude et la compréhension des exemples de réseaux de $PU(2,1)$construits par Mostow en 1980