One-relator surface groups

Extract

For X a subset of a group G, the smallest normal subgroup of G which contains X is called the normal closure of X and is denoted by ngp (X; G) or simply by ngp (X) if there is no possibility of ambiguity. By a surface group we mean the fundamental group of a compact surface. We are interested in determining when a normal subgroup of a surface group contains a simple loop – the homotopy class of an embedding of S¹ in the surface, or more generally, a power of a simple loop. This is significant to the study of 3-manifolds since a Heegaard splitting of a 3-manifold is reducible (cf. [2]) if and only if the kernel of the corresponding splitting homomorphism contains a simple loop. We give an answer in the case that the normal subgroup is the normal closure ngp (α) of a single element α: if ngp (α) contains a (power of a) simple loop β then α is homotopic to a (power of a) simple loop and β^±1 is homotopic either to (a power of) α or to the commutator [α, γ] of a with some simple loop γ meeting a transversely in a single point. This implies that if a is not homotopic to a power of a simple loop, then the quotient map π₁(S) → π₁(S)/ngp (α) does not factor through a group with more than one end. In the process we show that π₁(S)/ngp (α) is locally indicable if and only if α is not a proper power and that α always lifts to a simple loop in the covering space S_α of S corresponding to ngp (α). We also obtain some estimates on the minimal number of double points in certain homotopy classes of loops.