The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berthé et al.

We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in $\mathbb{C}^{n}, n \geq3$ , is not subelliptic.

We establish a parametric extension$h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from [12]. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.

We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of$g$copies of$S$^{$n$}×$S$^{$n$}, in the limit $${g \to \infty}$$ . Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.

In the moduli space $$ \mathcal{M} $$ _{$g$}of genus-$g$Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field$F$of degree$g$. If$g$= 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$ _{$g$}and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ . Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field$F$satisfying a strong geometry of numbers type restriction.