L-algebras and topology

[Formula: see text]-algebras are based on an equation which is fundamental in the construction of various torsion-free groups, including spherical Artin groups, Riesz groups, certain mapping class groups, para-unitary groups, and structure groups of set-theoretic solutions to the Yang–Baxter equation. A topological study of [Formula: see text]-algebras is initiated. A prime spectrum is associated to certain (possibly all) [Formula: see text]-algebras, including three classes of [Formula: see text]-algebras where the ideals are determined in a more explicite fashion. Known results on orthomodular lattices, Heyting algebras, or quantales are extended and revisited from an [Formula: see text]-algebraic perspective.

Naturality and Mapping Class Groups in Heegaard Floer Homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

Right-angled Artin groups as normal subgroups of mapping class groups

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.