Real Places and Surface Bundles

TL;DR

This paper constructs infinite families of hyperbolic surface bundles over the circle with specified punctures and Euler characteristics, whose trace fields have real places, supporting a broader conjecture about their existence.

Contribution

It provides new explicit examples of hyperbolic surface bundles with trace fields having real places, extending previous non-existence results.

Findings

Existence of infinite families of such bundles for various pairs (-χ, p)

Supports the conjecture that such examples exist for all pairs with finitely many exceptions

Extends known results by constructing explicit examples

Abstract

Our results complement D. Calegari's result that there are no hyperbolic once-punctured torus bundles over $S^1$ with trace field having real place. We exhibit several infinite families of pairs $(-\chi, p)$ such that there exist hyperbolic surface bundles with over $S^1$ with fiber having $p$ punctures and Euler characteristic $\chi$ with trace field of having a real place. This supports our conjecture that there exist such examples for each pair $(-\chi, p)$ with finitely many known exceptions.