Let H be a cyclically-presented group on n generators with a single defining relator. Attempts have been made to classify such groups by their order, their status as a 3-manifold group, and the asphericity status of their presentations. For groups with a defining relator of length 3 these classifications are nearly complete, with only two groups, H(9,4) and H(9,7), representing outstanding cases in each classification. We complete the asphericity classification for the presentations of these two groups and show that H(9,4) is not a 3-manifold group. We also determine that H(9,7) is a 3-manifold group if and only if it is cyclic of order 37.
We consider a relative presentation P for a natural degree-n split extension E of H, and apply a practical computational method to find reduced relative spherical pictures (a type of graph) over P. Our method uses a depth-first search to construct pictures region-by-region (i.e., face-by-face) from a pre-chosen starting region. New regions are typically added directly adjacent to the newest and oldest regions with available edges. This gives a construction of relative picture in a spiral ordering centered on the initial region. The addition of regions outside of this spiral ordering is sometimes required, but is done only in a limited capacity. Some user-defined limitations are also applied to prevent the search from continuing indefinitely down non-viable branches of the search graph. The method terminates when all edges have been connected--resulting in a complete picture--or when the search backs up and can no longer continue from the initial region.
We successfully apply our method to the split extensions arising from H(9,4) and H(9,7). In each case, the resulting symmetric picture reveals interesting relations in the group extension E. In particular, these relations can establish that the relative presentation P for E is not relatively aspherical, and hence the presentation for the cyclically-presented group H is not aspherical in each case. One of these two cyclically-presented groups, H(9,7), is also shown to contain a torsion element. The question of whether these groups are infinite remains unresolved.