We prove Torelli-type uniqueness theorems for both ${\mathrm{ALG}}^{\ast }$ gravitational instantons and ALG gravitational instantons which are of order 2. That is, the periods uniquely characterize these types of gravitational instantons up to diffeomorphism. We define a period mapping $\mathcal{P}$, which we show is surjective in the ALG cases, and has open image in the ${\mathrm{ALG}}^{\ast }$ cases. We also construct some new degenerations of hyper-Kähler metrics on the K3 surface which exhibit bubbling of ${\mathrm{ALG}}^{\ast }$ gravitational instantons.

For $g,n\ge 0$, we construct a 3-dimensional Calabi–Yau $A_{\mathrm{\infty }}$-category ${\mathcal{C}}_{g,n}$ such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}({\mathcal{C}}_{g,n})$, is a moduli space of quadratic differentials on a genus-g surface with simple zeros and n simple poles. For a generic point in the moduli space, we compute the corresponding quantum/refined Donaldson–Thomas (DT) invariants in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, we find that these counts satisfy wall-crossing formulas.

We prove that a $C^2$-generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.

Suppose a residually finite group G acts cocompactly on a contractible complex with strict fundamental domain Q, where the stabilizers are either trivial or have normal $\mathbb{Z}$-subgroups. Let $\partial Q$ be the subcomplex of Q with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of G. We show that independent of the chain, the normalized torsion limits to the torsion of $\partial Q$ shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual $L^2$-torsion of G in terms of the torsion of stabilizers and topology of $\partial Q$. In particular, we get complete answers for right-angled Artin groups, which shows that they satisfy a torsion analogue of Lück’s approximation theorem.