Group presentations corresponding to spines of $3$-manifolds. III

Abstract:

Continuing after the previous papers of this series, attention is devoted to RR-systems having two towns (i.e., to compact 3-manifolds with spines corresponding to group presentations having two generators). An interesting kind of symmetry is noted and then used to derive some useful results. Specifically, the following theorems are proved: Theorem 1. Let $\phi$ be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of $\phi$ involving just two generators a and b. If w is cyclically reduced, then either (a) w can be “written backwards" (i.e., if $w = {a^{{m_1}}}{b^{{m_1}}}{a^{{m_2}}}{b^{{n_2}}} \ldots {a^{{m_k}}}{b^{{n_k}}}$, then w is a cyclic conjugate of ${b^{{n_k}}}{a^{{m_k}}} \ldots {b^{{n_2}}}{a^{{m_2}}}{b^{{n_1}}}{a^{{m_1}}}$), or (b) w lies in the commutator subgroup of the free group on a and b. Theorem 2. (Loose translation). If $\phi$ is a group presentation with two generators and if the corresponding 2-complex ${K_\phi }$ is a spine of a closed orientable 3-manifold then, ${K_\phi }$ is a spine of a closed orientable 3-manifold if and (except for two minor cases) only if $\phi$ has two relators and among the six allowable types of syllables (3 in each generator), exactly four occur an odd number of times. Further, each of the two relators can be “written backwards."