Bounded geometry with no bounded pants decomposition

TL;DR

This paper constructs specific hyperbolic Riemann surfaces with bounded geometry that defy the existence of bounded pants decompositions, challenging previous assumptions in geometric topology.

Contribution

It provides explicit examples of hyperbolic surfaces with bounded geometry lacking bounded pants decompositions, expanding understanding of surface decompositions.

Findings

Existence of hyperbolic surfaces with bounded geometry but no bounded pants decomposition

Construction of quasiconformally homogeneous hyperbolic surfaces other than the hyperbolic plane

Counterexamples for bounded pants decompositions in infinite-type surfaces

Abstract

We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.