Representations of 3-manifold groups (Q2776346)

In this survey paper, the author focuses on the interaction between two kinds of representations of fundamental groups of compact, connected, orientable, irreducible \(3\)-manifolds: representations by \(2\times 2\) matrices (related to hyperbolic structures on 3-manifolds) and representations by automorphisms of trees (related to surfaces in 3-manifolds). NEWLINENEWLINENEWLINESection 1 shows how an essential surface in a 3-manifold \(M\) leads naturally to an action of \(\pi_1(M)\) on a tree \(T\) which is nontrivial (no vertex of \(T\) is fixed by the entire group \(\pi_1(M)\)) and without inversion (no element of \(\pi_1(M)\) fixes an edge of \(T\) exchanging its endpoints). NEWLINENEWLINENEWLINESection 2 presents a construction that goes the opposite way: starting with a nontrivial action of \(\pi_1(M)\) without inversion on a tree \(T\), an essential surface in \(M\) can be obtained. NEWLINENEWLINENEWLINESection 5 discusses when and how such an action can be constructed, using some concepts explained in Sections 3 and 4. NEWLINENEWLINENEWLINEIn the last subsection of Section 5 and in Section 6, an important application to topology is presented: the existence of an essential separating surface in the complement of a nontrivial knot, first conjectured by Neuwirth. NEWLINENEWLINENEWLINEA second application is given in Section 7, with a proof of the Generalized Smith Conjecture, which states that a cyclic covering of a connected, closed, orientable 3-manifold, branched over a nontrivial knot, cannot be simply connected. NEWLINENEWLINENEWLINEIn Section 8, the author defines a norm on \(H_1(\partial M,R)\) for hyperbolic 3-manifolds with connected torus boundary. This norm is used in Section 9 for applications to Dehn surgery and in Section 10 for studying more refined questions related to the Neuwirth Conjecture. NEWLINENEWLINENEWLINEFinally, in Section 11 is explained how the above techniques are related to geometric questions about degenerations of hyperbolic structures.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].