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We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex…

This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit geometric construction. Specifically, for each odd $d \ge 1$ there is a smooth projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth irreducible…

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the…

We prove that a cyclic cover of a smooth complex projective variety is Brody hyperbolic if its branch divisor is a generic small deformation of a large enough multiple of a Brody hyperbolic base-point-free ample divisor. We also show the…

For an affine double plane defined by an equation of the form z^2 = f, we study the divisor class group and the Brauer group. Two cases are considered. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a…

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed,…

We introduce a number of tools for finding and studying \emph{hierarchically hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups of surfaces, Teichm\"{u}ller space with either the Teichm\"{u}ller or…

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices…

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichm\"uller theory than arbitrary non-compact surfaces. We show that the…

Suppose that $X$ is an infinite, connected, locally finite, quasi-transitive graph with the property that every bi-infinite quasi-geodesic uniformly coarsely separates $X$ into exactly two deep pieces. We show that such an $X$ is…

In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new…

We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence,…

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for…

For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic…

We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface.

A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that, when all peripheral…

A separating ($M-2$)-curve is a smooth geometrically irreducible real projective curve $X$ such that $X(\mathbb{R})$ has $g-1$ connected components and $X(\mathbb{C})\setminus X(\mathbb{R})$ is disconnected. Let $T_g$ be a Teichm\"uller…