GROUP PRESENTATIONS CORRESPONDING TO SPINES OF 3 - MANIFOLDS . Ill

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 tf> be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of $ involving just two generators a and b.Ifwis cyclically reduced, then either (a) w can be "written backwards" (i.e., if w = am'bm'amibni... amkbHt, then w is a cyclic conjugate of ¿>"*a"* ... b"2am,b''>am>), or (b) w lies in the commutator subgroup of the free group on a and b. Theorem 2. (Loose translation). If is a group presentation with two generators and if the corresponding 2-complex K^ is a spine of a closed orientable 3-manifold lhen,K^ is a spine of a closed orientable 3-manifold if and (except for two minor cases) only if «¡> 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." In [3] RR-systems were introduced and were shown to contain much useful information about the relationship between group presentations and spines of orientable 3-manifolds. In this paper we devote our attention to RR-systems having two towns and correspondingly to group presentations with two generators. We observe that many of these RR-systems have a certain kind of symmetry. By means of this symmetry we will derive some useful results. In particular, we will establish the following: Theorem 1. Let <p be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of <p involving just two generators a and b. If w is cyclically reduced, then w can be "written backwards" or lies in the commutator subgroup of the free group on a and b. (Specifically, w can be "written backwards" ifw = am,b"'am2b''2. .. a^b* is a cyclic conjugate qfw = b^aTM*... b"2am2bn'ami.) (Cf. [5].) Received by the editors April 20, 1976. AMS (MOS) subject classifications (1970). Primary 57A10; Secondary 55A25.