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

Abstract

Let $M_{\varphi }$ be a surface bundle over a circle with monodromy $\varphi :S\to S$. We study deformations of certain reducible representations of ${\pi }_1(M_{\varphi })$ into $\text{SL}(n,\mathbb{C})$, obtained by composing a reducible representation into $\text{SL}(2,\mathbb{C})$ with the irreducible representation $\text{SL}(2,\mathbb{C})\to \text{SL}(n,\mathbb{C})$. In particular, we show that under certain conditions on the eigenvalues of ${\varphi }^{\ast }$, 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 $\partial M_{\varphi }$. 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.

Corrections to some arguments in Sections 3 & 4 and various typographical fixes. Corrected and updated main arguments using more direct calculations. 14 pages, 1 figure. Accepted by Osaka J. Math