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

Abstract

Let $M_ϕ$ be a surface bundle over a circle with monodromy $ϕ:S \rightarrow S$. We study deformations of certain reducible representations of $π_1(M_ϕ)$ 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 $ϕ^*$, 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 $\partial M_ϕ$. 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.