On the Hyperbolicity of Small Cancellation Groups and One-relator Groups

In the article, a result relating to maps (= finite planar connected and simply connected 2-complexes) that satisfy a C(p)&T (q) condition (where (p, q) is one of (3, 6), (4, 4), (6, 3) which correspond to regular tessellations of the plane by triangles, squares, hexagons, respectively) is proven. On the base of this result a criterion for the Gromov hyperbolicity of finitely presented small cancellation groups satisfying non-metric C(p)&T (q)-conditions is obtained and a complete (and explicit) description of hyperbolic groups in some classes of one-relator groups is given: All one-relator hyperbolic groups with > 0 and ≤ 3 occurrences of a letter are indicated; it is shown that a finitely generated onerelator group G whose reduced relator R is of the form R ≡ aT0aT1 . . . aTn−1, where the words Ti are distinct and have no occurrences of the letter a ±1, is not hyperbolic if and only if one has in the free group that (1) n = 2 and T0T −1 1 is a proper power; (2) n = 3 and for some i it is true (with subscripts mod 3) that TiT −1 i+1TiT −1 i+2 = 1; (3) n = 4 and for some i it is true (with subscripts mod 4) that TiT −1 i+1Ti+2T −1 i+3 = 1.