Related papers: McShane's identity for classical Schottky Groups
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have \[ \sum_{\gamma} \frac{1}{e^{\ell(\gamma)}+1}={1/2} \] where $\gamma$ varies over the homotopy classes of essential…
Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these…
Real points of Schottky space ${\mathcal S}_{g}$ are in correspondence with extended Kleinian groups $K$ containing, as a normal subgroup, a Schottky group $\Gamma$ of rank $g$ such that $K/\Gamma \cong {\mathbb Z}_{2n}$ for a suitable…
In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms…
In this paper, we introduce the discrete conformal structures on surfaces with boundary in an axiomatic approach, which ensures that the Poincar\'{e} dual of an ideally triangulated surface with boundary has a good geometric structure.Then…
Locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^2$ were classified by E.\,Cartan in 1932. In this work, we complete the classification of locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$.
In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the…
Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichm\"uller spaces, and hyperbolic 3-manifolds. In the same paper, they…
We propose a definition for the length of closed geodesics in a globally hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closed geodesics of length less than $R$ grows exponentially fast with $R$…
The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing…
We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly…
Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group.…
Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of…
We extend results of Bhagwat and Rajan on a strong multiplicity one property for length spectrum to hyperbolic manifolds with cusps, showing that for two even dimensional hyperbolic manifolds of finite volume, if all but finitely many…
The trace set of a Fuchsian group $\Gamma$ ist the set of length of closed geodesics in the surface $\Gamma \backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded…
This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into…
We introduce a Grothendieck group $E_n$ for bounded polytopes in $\mathbb R^n$. It differs from the usual Euclidean scissors congruence group in that lower-dimensional polytopes are not ignored. We also define an analogous group $L_n$ using…
We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary…
We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in…