Uniform spectral gap of scl in $2$-orbifolds

TL;DR

This paper establishes a uniform spectral gap for stable commutator length in hyperbolic 2-orbifolds, with explicit bounds, aiding the study of 3-manifolds, using quasimorphisms and hyperbolic geometry.

Contribution

It provides explicit uniform spectral gap estimates for stable commutator length in hyperbolic 2-orbifolds, including special cases, advancing understanding in 3-manifold topology.

Findings

Uniform spectral gap of 1/36 for most hyperbolic 2-orbifolds

Explicit quasimorphisms constructed for generic cases

Hyperbolic geometry techniques applied to exceptional cases

Abstract

We show a uniform spectral gap of stable commutator length for all compact hyperbolic $2$-orbifolds relative to the peripheral subgroups. Except for the case of a sphere with three cone points, we have an explicit uniform gap $1/36$. These estimates are needed in understanding stable commutator length in $3$-manifolds. Our methods use explicit quasimorphisms for the generic case, and use hyperbolic geometry (pleated surfaces) for the exceptional case of a sphere with three cone points.