Hodge-Teichmueller planes and finiteness results for Teichmueller curves

TL;DR

This paper establishes finiteness results for algebraically primitive Teichmueller curves in high genus strata, introduces Hodge-Teichmueller planes as a key tool, and shows these curves are not dense, with implications for non-arithmetic curves.

Contribution

It introduces Hodge-Teichmueller planes and proves finiteness and non-density of algebraically primitive Teichmueller curves in certain strata and genus levels.

Findings

Finitely many algebraically primitive Teichmueller curves in each prime genus ≥ 3.

Algebraically primitive Teichmueller curves are not dense in any stratum component.

Finiteness of non-arithmetic Teichmueller curves in specific hyperelliptic strata.

Abstract

We prove that there are only finitely many algebraically primitive Teichmueller curves in the minimal stratum in each prime genus at least 3. The proof is based on the study of certain special planes in the first cohomology of a translation surface which we call Hodge-Teichmueller planes. We also show that algebraically primitive Teichmueller curves are not dense in any connected component of any stratum in genus at least 3; the closure of the union of all such curves (in a fixed stratum) is equal to a finite union of affine invariant submanifolds with unlikely properties. Results of this type hold even without the assumption of algebraic primitivity. Combined with work of Nguyen and the second author, a corollary of our results is that there are at most finitely many non-arithmetic Teichmueller curves in H(4)^hyp.