Related papers: On Thin Heckoid and Generalised Triangle Groups in…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. We describe and classify finite, flat, and linearly reductive subgroup schemes of $\mathrm{SL}_2$ over $\mathrm{Spec}\:\mathcal{O}_K$. We also establish finiteness results for…

In this article, we single out representations of surface groups into $\mathsf{PSL}_d(\mathbb{C})$ which generalize the well-studied family of pleated surfaces into $\mathsf{PSL}_2(\mathbb{C})$. Our representations arise as sufficiently…

Let G be a connected, compact, semisimple algebraic group over the field of real numbers R. Using Kac diagrams, we describe combinatorially the first Galois cohomology sets H^1(R,H) for all inner forms H of G. As examples, we compute…

We introduce the notion of finite slope families to encode the local properties of the p-adic families of Galois representations appearing in the work of Harris, Lan, Taylor and Thorne on the construction of Galois representations for…

We extend the computations in [AGM1, AGM2, AGM3] to find the cohomology in degree five of a congruence subgroup Gamma of SL(4,Z) with coefficients in a field K, twisted by a nebentype character eta, along with the action of the Hecke…

Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.

We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and…

Our work is based on the stacked weak-lensing analysis of a sample of 80 strong-lensing galaxy groups. Our main results are the following: (i) the lensing signal does not allow us to firmly reject a simple singular isothermal sphere mass…

We present a partial classification of those finite linear spaces $\mathcal{S}$ on which an almost simple group $G$ with socle $PSL(3,q)$ acts line-transitively.

Packings of regular convex polygons ($n$-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly…

Let $G$ be a split real form of a complex simple adjoint group whose Weyl group contains $-1$, let $\lambda$ be the Jordan projection of $G$, and let $S$ be a closed orientable surface of genus at least 2. For a $G$-Hitchin representation…

This paper is devoted to the study of tessellations of the hyperbolic plane, especially the ones associated to hyperbolic triangle groups $\Delta(l,m,n)$. We give a full description of the cone types of these graphs and show that their…

We investigate a new family of regions that is the universal generalization of three well-known region families in the field of enumeration of tilings: the quasi-regular hexagons, the semi-hexagons, and the halved hexagons. We prove a…

The goal of this paper is to give a conjectural census of complex hyperbolic sporadic groups. We prove that only finitely many of these sporadic groups are lattices. We also give a conjectural list of all lattices among sporadic groups, and…

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma<G$ of higher rank…

We determine the growth of the dimension of the slope subspaces of the cohomology of arithmetic subgroups in reductive algebraic groups as a function of the slope.

We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$,…

We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an…

We give results on when a finitely generated group has only indiscrete embeddings in SL(2,C), with particular reference to 3-manifold groups. For instance if we glue two copies of the figure 8 knot along its torus boundary then the…

We classify twisted conjugacy classes of type D associated to the sporadic simple groups. This is an important step in the program of the classification of finite-dimensional pointed Hopf algebras with non-abelian coradical. As a by-product…