A note on the existence of essential tribranched surfaces

TL;DR

This paper demonstrates that any 3-manifold with a fundamental group of rank four or higher contains an essential tribranched surface, extending the understanding of surface embeddings in 3-manifolds.

Contribution

It proves the existence of essential tribranched surfaces in 3-manifolds with fundamental groups of rank four or more, generalizing previous concepts.

Findings

Existence of essential tribranched surfaces in certain 3-manifolds

Extension of surface embedding theory in 3-manifolds

Fundamental group rank condition for existence

Abstract

The second author and Hara introduced the notion of an essential tribranched surface that is a generalisation of the notion of an essential embedded surface in a 3-manifold. We show that any 3-manifold for which the fundamental group has at least rank four admits an essential tribranched surface.