Related papers: CAT(-1)-Type Properties for Teichm\"uller Space

We prove an explicit equivalence between various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, we prove that for X a CAT(0) space and $\gamma$ a quasi-geodesic, the following four statements are equivalent…

Following Beck-Chen, we say a flow $\phi_t$ on a metric space $(X, d)$ is superdense if there is a $c > 0$ such that for every $x \in X$, and every $T>0$, the trajectory $\{\phi_t x\}_{0 \le t \le cT}$ is $1/T$-dense in $X$. We show that a…

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a…

We classify the Teichm\"uller curves in the moduli space of genus three Riemann surfaces $\mathcal M_3$ that are obtained by a covering construction from a primitive Teichm\"uller curve in $\mathcal M_2$. We describe the action on homology…

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of…

In this century, a square-tiled translation surface (an origami) is intensively studied as an object with special properties of its translation structure and its $SL(2,\mathbb{R})$-orbit embedded in the moduli space. We generalize this…

A main goal in lattice theory is the construction of dense lattices. Most of the remarkable dense lattices in small dimensions have an additional symmetry, they are modular, i.e. similar to their dual lattice. Extremal lattices are densest…

We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $\gamma$ in such a torus is said to be homologically maximizing if one (hence every) lift of $\gamma$ to the universal cover is arclength…

We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are derived from the parametrisation of the moduli spaces by the bundle of measured geodesic…

In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichm\"uller geodesic trajectories.The main tool is quantitative nondivergence of…

We study the wall-crossing for moduli spaces of coherent systems of dimension one and order one on a smooth projective variety over the complex numbers. We compute the topological Euler characteristic of the moduli spaces in the particular…

Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show,…

We compute the Euler characteristics of the recently discovered series of Gothic Teichm\"{u}ller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichm\"{u}ller curves. Contrary…

We study modularity of the lattice Lat $(T)$ of closed invariant subspaces for a $C_0$-operator $T$ and find a condition such that Lat $(T)$ is a modular. Furthermore, we provide a quasiaffinity preserving modularity.

We prove that the image of the lifted period map on the universal cover lies in a complex Euclidean space. We also prove that the Teichm\"uller spaces of a class of polarized manifolds have complex affine structures.

Lipman Bers' universal Teichm\"uller space, classically denoted by $T(1)$, plays a significant role in Teichm\"uller theory, because all the Teichm\"uller spaces $T(G)$ of Fuchsian groups $G$ can be embedded into it as complex submanifolds.…

We derive new closed form expressions for the partition functions of free conformally-coupled scalars on $S^{2D-1}\times S^1$ which resum the exact high-temperature expansion. The derivation relies on an identification of the partition…

Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained…

In an infinite-dimensional Teichm\"uller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichm\"uller space $AT(\Delta)$. We…

For a compact surface $X_0$, Thurston introduced a compactification of its Teichm\"uller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a…