The dimension of Thurston's spine

TL;DR

This paper demonstrates that for sufficiently large genus, the set of hyperbolic surfaces with filling systoles forms a high-dimensional subset of moduli space, exceeding the expected cohomological dimension.

Contribution

It establishes a lower bound on the dimension of Thurston's proposed spine in moduli space, showing it surpasses the virtual cohomological dimension for large genus.

Findings

Dimension of the set of surfaces with filling systoles grows linearly with genus

For large genus, this set's dimension exceeds the virtual cohomological dimension

Existence of surfaces with filling systoles in high-dimensional subsets of moduli space

Abstract

We show that for every $\varepsilon>0$, there exists some $g\geq 2$ such that the set of closed hyperbolic surfaces of genus $g$ whose systoles fill has dimension at least $(5-\varepsilon) g$. In particular, the dimension of this set -- proposed as a spine for moduli space by Thurston -- is larger than the virtual cohomological dimension of the mapping class group.