Deformations of reducible SL(n,C) representations of fibered 3-manifold groups

TL;DR

This paper investigates the deformation space of certain reducible SL(n,C) representations of fibered 3-manifold groups, revealing conditions under which these representations lie in high-dimensional components and are limits of irreducible representations.

Contribution

It characterizes the deformation space of reducible SL(n,C) representations for fibered 3-manifolds, extending understanding of their structure and limits.

Findings

Reducible representations lie in high-dimensional components under specific eigenvalue conditions.

The results apply to mapping tori of pseudo-Anosov maps with certain eigenvalue restrictions.

Reducible representations can be limits of irreducible representations in the deformation space.

Abstract

Let $M_\phi$ be a surface bundle over a circle with monodromy $\phi:S \rightarrow S$. We study deformations of certain reducible representations of $\pi_1(M_\phi)$ into $\text{SL}(n,\mathbb{C})$, obtained by composing a reducible representation into $\text{SL}(2,\mathbb{C})$ with the irreducible representation $\text{SL}(2,\mathbb{C}) \rightarrow \text{SL}(n,\mathbb{C})$. In particular, we show that under certain conditions on the eigenvalues of $\phi^*$, the reducible representation is contained in a $(n+1+k)(n-1)$ dimensional component of the representation variety, where $k$ is the number of components of $\partial M_\phi$. This result applies to mapping tori of pseudo-Anosov maps with orientable invariant foliations whenever 1 is not an eigenvalue of the induced map on homology, where the reducible representation is also a limit of irreducible representations.