Bers' constants for punctured spheres and hyperelliptic surfaces

TL;DR

This paper proves Buser's conjecture that hyperbolic spheres with cusps have pants decompositions with geodesics bounded by roughly the square root of the number of cusps, advancing understanding of geometric decompositions.

Contribution

It establishes new bounds for Bers' constants on punctured spheres and hyperelliptic surfaces, confirming a conjecture and providing bounds for various surface types.

Findings

Hyperbolic spheres with n cusps have pants decompositions with geodesics bounded by roughly √n.

Provides lower and upper bounds for Bers' constants on hyperelliptic surfaces.

Confirms Buser's conjecture for spheres with cusps.

Abstract

This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with $n$ cusps has a pants decomposition with all of its geodesics of length bounded by a constant roughly square root of $n$. Other results include lower and upper bounds for Bers' constants for hyperelliptic surfaces and spheres with boundary geodesics.