Sirio Resteghini | Searcharxiv

arXiv2025

Simultaneous Sequential Compactness

A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $\omega$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or in different sequentially compact spaces; we link the results for different spaces to ones for the same space; we show that simultaneous convergence happens for less than $\mathfrak s$ sequences in spaces with weight bounded by $\mathfrak s$ and for less than $\mathfrak h$ sequences in general; we show a slight generalisation of these results in the context of Hausdorff spaces; and finally we investigate their optimality.

math.GN↗

arXiv2024

Free Products and the Isomorphism between Standard and Dual Artin Groups

Given a Coxeter system with a fixed Coxeter element, there is a surjective group morphism $\Psi$ from the standard to the dual Artin groups. We give conditions that are sufficient, necessary or equivalent to $\Psi$ being an isomorphism. In particular, we prove that if the Hurwitz action on the reduced words of any element in the noncrossing partition poset is transitive, and if the Hurwitz action on the reduced words of the Coxeter element has the same stabilizer as essentially the same action viewed in the standard Artin group, then $\Psi$ is an isomorphism. Both of those conditions are already known in some cases, notably in spherical and affine types. We then prove that taking the free (or direct) product of groups that satisfy those two conditions yields another group that, with a suitable Coxeter element, also satisfies them.

math.GR↗