Detection of essential surfaces in 3-manifolds with $SL_2$-trees

Abstract

According to Bass-Serre theory [Se], given a field F with a discrete valuation v : F ∗→Z, there is a canonical way to construct a simplicial tree T = TF,v on which SL2(F ) acts simplicially without inversion. When a group π has a representation into the group SL2(F ), then there is an induced action of π on the tree via the representation. If further π is the fundamental group of a compact 3-manifoldM and the action ofπ on the tree T is nontrivial (meaning that there is no point on T fixed by every element of π ), then the action induces a splitting of M along an essential surface in the following way: let M p −→ M be the universal covering, there is a π1(M)-equivariant map f : M→T which is transverse to the set E of midpoints of edges in T such that p(f −1(E)) is an essential surface in M .We say such an essential surface is associated to (or detected by) an SL2-tree. A natural question is: which essential surfaces in a compact 3-manifold can be associated to SL2-trees and how do they depend on the choice of the field and discrete valuation? This is the main issue we are going to address in this paper. Any 3-manifold mentioned in this paper is automatically assumed orientable and connected. By an essential surface in a compact 3-manifold M we mean an orientable, properly embedded, incompressible surface each component of which is neither boundary parallel in M nor bounds a 3-ball (when it is a 2sphere). Also recall that a discrete valuation on a field F is a homomorphism v from the multiplicative group F ∗ onto the group of integers Z such that v(a + b) ≥ min(v(a), v(b)) for any a, b ∈ F ∗ with a + b = 0. Let v : F ∗→Z be a discrete valuation. The set Ov = {0} ∪ {a ∈ F ∗; v(a) ≥ 0} S. Schanuel · X. Zhang Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, USA (e-mail: schanuel@buffalo.edu / xinzhang@math.buffalo.edu)