Tropical moduli spaces as symmetric Δ$\Delta$‐complexes

Bulletin of the London Mathematical Society · 2022 · 12 citations

• Computer Science
• Library science
• Mathematics

We develop techniques for studying fundamental groups and integral singular homology of symmetric Δ $\Delta$ -complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Δ g $\Delta _g$ and Δ g , n $\Delta _{g,n}$ are simply connected, for g ⩾ 1 $g \geqslant 1$ . We also show that Δ 3 $\Delta _3$ is homotopy equivalent to the 5-sphere, and that Δ 4 $\Delta _4$ has 3-torsion in H 5 $H_5$ .

Most big mapping class groups fail the Tits alternative

arXiv (Cornell University) · 2021 · 1 citations

• Computer Science
• Artificial Intelligence
• Mathematics

Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits Alternative. That is, Map$(X)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.