Related papers: Tropical limit of hyperbolic amoebas of complex an…

We introduce a type of minimal surface in the pseudo-hyperbolic space $\mathbb{H}^{n,n}$ (with $n$ even) or $\mathbb{H}^{n+1,n-1}$ (with $n$ odd) associated to cyclic $\mathrm{SO}_0(n,n+1)$-Higg bundles. By establishing the infinitesimal…

We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv,…

We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.

For any abelian category \calC satsifying (AB5) over a separated, quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the flat site of S. We will give a concrete description of \GL(\calC) when \calC is the category of…

In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of hyperplane stabilizers with…

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a…

The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree…

We prove that every discrete faithful representation of the surfcae group into SL(2,C) is the monodromy of a holomorphic connection on the trivial rank-2 vector bundle over a Riemann surface. As an application, we answer the question posed…

We prove that for a one-ended hyperbolic graph $X$, the size of the quotient $X/G$ by a group $G$ acting freely and cocompactly bounds from below the number of simplices in an Eilenberg-MacLane space for $G$. We apply this theorem to show…

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known…

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively…

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant…

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega_{X} \otimes…

The restriction of a supercuspidal representation of SL_2(k), for k a local nonarchimedean field, to a maximal compact subgroup decomposes as a multiplicity-free direct sum of irreducible representations. We explicitly describe this…

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective…

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}:…

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal…

We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem…

The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and…

For any irreducible representation of a surface group into $\mathrm{SL}_2(\mathbb{C})$, we show that there exists a pants decomposition where the restriction to any pair of pants is irreducible and where no curve of the decomposition is…