We prove a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin and Sullivan’s theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of discrete conformality. The proof follows the same strategy that Rodin and Sullivan used by establishing a rigidity result for regular hexagonal triangulations of the plane and estimating the quasiconformal constants associated to the discrete conformal maps.

Inspired by the log Gromov–Witten (or GW) theory of Gross–Siebert/Abramovich–Chen, we introduce a geometric notion of log $J$–holomorphic curve relative to a simple normal crossings symplectic divisor defined by Tehrani–McLean–Zinger (2018). Every such moduli space is characterized by a second homology class, genus and contact data. For certain almost complex structures, we show that the moduli space of stable log $J$–holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (ie the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In an earlier preprint, we introduced a natural set up for studying the deformation theory of log (and relative) curves and obtained a logarithmic analogue of the space of Ruan–Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space, and define relative GW invariants without any restriction.

We prove explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic $3$–manifold $N$ and the thick part of any of its long Dehn fillings. Given a bilipschitz constant $J>1$ and a thickness constant $𝜖>0$, we quantify how long a Dehn filling suffices to guarantee a $J$–bilipschitz map on $𝜖$–thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff’s theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation.

Our quantitative results have two immediate applications. First, we relate the Margulis number of $N$ to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed $3$–manifold $M$ whose Margulis number is less than $0.29$. Combined with Shalen’s upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of $3$–manifolds whose Margulis numbers are below $0.29$.

Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold $N$, we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on $N$. This reduces the cosmetic surgery conjecture on $N$ to an explicit finite search.

We construct a family of infinite simple groups that we call twisted Brin–Thompson groups, generalizing Brin’s higher-dimensional Thompson groups $sV$ for $s\in \mathbb{N}$. We use twisted Brin–Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $sV$ and hence every right-angled Artin group, including examples of type ${\text{F}}_{\infty }$ and a family of examples of type ${\text{F}}_{n-1}$ but not of type ${\text{F}}_n$ for arbitrary $n\in \mathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.

We extend Lawrence’s representations of the braid groups to relative homology modules and we show that they are free modules over a ring of Laurent polynomials. We define homological operators and we show that they actually provide a representation for an integral version for $U_q\mathfrak{𝔰}\mathfrak{𝔩}(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules and we show it preserves the integral ring of coefficients, the action of $U_q\mathfrak{𝔰}\mathfrak{𝔩}(2)$, the braid group representation and its grading. This recovers an integral version for Kohno’s theorem relating absolute Lawrence representations with the quantum braid representation on highest-weight vectors. This is an extension of the latter theorem as we get rid of generic conditions on parameters, and as we recover the entire product of Verma modules as a braid group and a $U_q\mathfrak{𝔰}\mathfrak{𝔩}(2)$–module.

Given a connected, oriented, complete, finite-area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of simple closed multigeodesics on $X$ of a prescribed topological type and total hyperbolic length $\le L$ is asymptotic to a polynomial in $L$ of degree $6g-6+2n$ as $L\to \infty$. We establish asymptotics of the same kind for counts of simple closed multigeodesics that keep track of the hyperbolic length of individual components rather than just the total hyperbolic length, proving a conjecture of Wolpert. The leading terms of these asymptotics are related to limits of Weil–Petersson volumes of expanding subsets of quotients of Teichmüller space. We introduce a framework for computing limits of this kind in terms of purely topological information. We provide two further applications of this framework to counts of square-tiled surfaces and counts of filling closed multigeodesics on hyperbolic surfaces.

We prove that for a compact $3$–manifold $M$ with boundary admitting an ideal triangulation $\mathcal{𝒯}$ with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that $\mathcal{𝒯}$ is isotopic to a geometric decomposition of $M$. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo (Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 12–20) for pseudo-$3$–manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.

We study the asymptotically rigid mapping class groups of infinitely punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy–Thompson groups $T^{♯},T^{\ast }$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}{H}_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube stabilizers isomorphic to finite extensions of braid groups. As an application, we prove conjectures of Funar–Kapoudjian and Degenhardt by showing that $T^{♯}$ and $T^{\ast }$ are of type $F_{\infty }$ and that $\mathrm{br}{H}_n$ is of type $F_{n-1}$ but not of type $F_n$.