We study weak approximation and the Hilbert property for Campana points, both of importance in recent work on a Manin-type conjecture by Pieropan, Smeets, Tanimoto, and Várilly-Alvarado. We show that weak weak approximation implies the Hilbert property for Campana points, and we exploit this to exhibit Campana orbifolds whose sets of Campana points are not thin.

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective.

We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding.

Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a ${\mathit{C}}^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with ${\mathit{C}}^2$ boundary $\mathit{D}\subset {\mathbb{C}}^{\mathit{n}}$, we show the existence of global holomorphic coordinate systems defined on $\bar{\mathit{D}}$ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small ${\mathit{C}}^2$ perturbation of $\partial \mathit{D}$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a ${\mathit{C}}^2$ real hypersurface $\mathit{M}\subset {\mathbb{C}}^{\mathit{n}}$, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.

In response to W. Kuperberg’s question, for every dimension at least two, there are uncountably many compact metric spaces, each of which (1) has the shape of the sphere of that dimension and (2) contains an infinite sequence of pairwise disjoint closed subsets irreducibly having nonzero top-dimensional integer Čech–Alexander–Spanier cohomology.

The Friedman–Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple $(\mathit{G},\mathit{X},\mathit{H})$ consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H that is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of $\mathrm{Out}({\mathit{F}}_{\mathit{n}})$, and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich–Rafi and Dowdall–Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.

We study trisections of smooth compact nonorientable 4-manifolds and introduce trisections of nonorientable 4-manifolds with boundary. In particular, we prove a nonorientable analogue of a classical theorem of Laudenbach–Poénaru and analogues for some well-known theorems from 3-manifold topology. As a consequence, trisection diagrams and Kirby diagrams of closed nonorientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of nonorientable 4-manifolds with many examples.