# Invariant incompressible surfaces in reducible 3-manifolds

## TL;DR

This paper investigates how the mapping class group influences invariant incompressible surfaces in reducible 3-manifolds and characterizes when such manifolds admit Anosov tori.

## Contribution

It provides a classification of reducible 3-manifolds admitting Anosov tori based on their prime summands, answering a specific open question.

## Key findings

- A reducible 3-manifold admits an Anosov torus iff a prime summand is the 3-torus, the mapping torus of -id, or a hyperbolic automorphism.
- The study links the mapping class group action to the existence of special invariant surfaces.
- The paper characterizes the structure of invariant incompressible surfaces under self-homeomorphisms.

## Abstract

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-id$, or the mapping torus of a hyperbolic automorphism.