ON THE HYPERBOLICITY OF SMALL CANCELLATION GROUPS AND ONE-RELATOR GROUPS

In the article, a result relating to maps (= finite planar connected a nd simply connected 2-complexes) that satisfy a C(p)&T(q) condition (w here (p, q) is one of (3, 6), (4; 4), (6, 3) which correspond to regul ar tessellations of the plane by triangles, squares, hexagons, respect ively) is proven. On the base of this result a criterion for the Gromo v hyperbolicity of finitely presented small cancellation groups satisf ying non-metric C(p)&T(q)-conditions is obtained and a complete (and e xplicit) description of hyperbolic groups in some classes of one-relat or groups is given: All one-relator hyperbolic groups with > 0 and les s than or equal to 3 occurrences of a letter are indicated; it is show n that a finitely generated one-relator group G whose reduced relator R is of the form R=aT(0)aT(1)...aT(n-1), where the words T-i are disti nct and have no occurrences of the letter a(+/-1), is not hyperbolic i f and only if one has in the free group that (1) n = 2 and T0T1-1 is a proper power; (2) n = 3 and for some i it is true (with subscripts mo d3) that TiTi+1-1TiTi+2-1 = 1; (3) n = 4 and for some i it is true (wi th subscripts mod 4) that TiTi+1-1Tz+2Ti+3-1 = 1.