We study the problem of existence of a periodic point in the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carathéodory’s prime ends rotation number, similar to Poincaré’s theory for circle homeomorphisms. In particular, we prove the converse of a classic result of Cartwright and Littlewood. The results are proved in a general context for homeomorphisms of arbitrary surfaces with a weak nonwandering-type hypothesis, which allows for applications in several different settings. The most important consequences are in the $C^r$-generic area-preserving context, building on previous work of Mather.

Let $\Gamma <{PSL}_2\left(\mathbb{R}\right)$ be a lattice and let $\omega \in \mathbb{H}$ be a point in the upper half-plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between geodesic rays of the lattice $\Gamma \omega$ intersected with increasingly large balls centered at $\omega$, thus proving a conjecture by Boca, Popa, and Zaharescu.

We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmüller distance for random walks on mapping class groups and on Cayley graphs of a large class of finitely generated groups.

We give a sharp lower bound for the self-intersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the generalized Clifford–Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $vol\left(X\right)\ge 2n!\chi \left({\omega }_X\right)$ and it is sharp.

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/\Gamma$ where $X$ is a contractible Riemannian manifold and $\Gamma <Isom\left(X\right)$ is a discrete subgroup, typically with infinite covolume. The existence depends on the $L^2$-Betti numbers of $\Gamma$, its subgroups, and a uniform lattice of $Isom\left(X\right)$. As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form ${\mathbb{H}}^4/\Gamma$ where ${\mathbb{H}}^4$ is real hyperbolic $4$-space and $\Gamma <Isom\left({\mathbb{H}}^4\right)$ is discrete and isomorphic to a subgroup of the fundamental group of a complete finite-volume hyperbolic $3$-manifold. Via Patterson–Sullivan theory, this implies the existence of a uniform positive upper bound on the Hausdorff dimension of the conical limit set of such a $\Gamma$ when $\Gamma$ is geometrically finite. Another application shows the existence of a uniform positive lower bound on the zeroth eigenvalue of the Laplacian of ${\mathbb{H}}^n/\Gamma$ over all discrete free groups $\Gamma <Isom\left({\mathbb{H}}^n\right)$ whenever $n\ge 4$ is even. (The bound depends on $n$.) This extends results of Phillips–Sarnak and Doyle, who obtained such bounds for $n\ge 3$ when $\Gamma$ is a finitely generated Schottky group.