Geodesic edge paths in nonpositively curved complexes

AbstractThis paper concerns three classes of geometric 2-complexes of nonpositive curvature: one in which all of the 2-cells are squares, one in which they are all equilateral triangles, and one in which they are all regular hexagons. (These cases correspond to the three regular tessellations of the euclidean plane.)These three classes of 2-complexes, while highly restrictive, are nevertheless useful because they include the Cayley complexes of group presentations satisfying the small cancellation conditions C″(p)−T(q) for (p,q)∈{(3,6),(4,4),(6,3)} (and satisfying the additional condition that all the relators have length exactly p). These three cases are of particular interest because such groups are not necessarily word-hyperbolic in the sense of Gromov (all other classes of C″(p)−T(q) groups are, see S.M. Gersten, H. Short, Small cancellation theory and automatic groups, Invent. Math. 102 (1990) 305–334). Moreover, they include presentations of alternating link groups (under free product with the infinite cyclic group; see C. Weinbaum, The word and conjugacy problem for the knot group of any tame prime alternating knot, Proc. Math. Soc. 22 (1971) 22–26).This paper is a study of geodesic edge paths in the geometric complexes described above, and consequently, of geodesic words in C″(p)−T(q) group presentations. The most fundamental theorem on this topic, namely that an edge path in the complex (or equivalently, a word on the group generators) that fails to be geodesic can be replaced via an efficient algorithm with an equivalent one that is geodesic, is well-known; but careful proofs of this fact, as well as related facts, are difficult or impossible to find in the literature. This paper is intended to provide rigorous proofs of some useful facts about the geodesics in these complexes and group presentations. Some embedding and minimality sufficiency conditions for disks are shown, together with an algorithm for finding all geodesics in an equivalence class. Some useful implications are mentioned K. Johnsgard, Two automatic spanning trees in small cancellation group presentations, Int. J. Algebra. Comput. 6 (1994) 429–440, K. Johnsgard, The conjugacy problem for groups of alternating prime tame links is polynomial-time, Trans. Amer. Math. Soc. 349 (1997) 847–901