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

We determine necessary and sufficient conditions for real algebraic curves near the non-singular tropical limit to be hyperbolic with respect to a point, thus generalising Speyer's classification of stable curves near the tropical limit. In…

In this paper we develop the theory of approximation for holomorphic null curves in the special linear group ${\rm SL}_2(\mathbb{C})$. In particular, we establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for the family…

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of $\operatorname{SL}(n)$ in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and…

A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.

Let K be a fine hyperbolic graph and G be a group acting on K with finite quotient. We prove that G is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups…

We prove the congruence subgroup property for the centralizer of a finite subgroup $G$ in the mapping class group of a hyperbolic oriented and connected surface of finite topological type $S$ such that the genus of the quotient surface…

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They…

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In…

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic…

\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we…

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups…

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face…

For two-scale homogenization of a general class of asymptotically degenerating %uniformly strongly elliptic symmetric PDE systems with a critically scaled high contrast in periodic coefficients of a small period $\varepsilon$, we derive a…

We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex…

We prove a bound conjectured by Itenberg on the Betti numbers of real algebraic hypersurfaces near non-singular tropical limits. These bounds are given in terms of the Hodge numbers of the complexification. To prove the conjecture we…

It is known that the hyperbolic plane admits a countable infinity of compactifications into a closed disk such that the isometric action of SL(2;R) acts analytically on the compactified space. We prove that among those compactifications,…

This paper is a more succinct version of the author's 1993 UCLA mathematics thesis. It proves that any group quasi-isometric to the product of the hyperbolic plane with the real line is a finite extension of a cocompact lattice in either…

Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid $(s^0)^2-(s^1)^2+(s^2)^2-(s^3)^2=1$ are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space…

We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an…

We find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic C*-dynamical system to be a maximal norm-closed subalgebra. Our conditions are in terms of the Arveson spectrum of the action. We…