On the hyperbolicity of small cancellation groups and one-relator groups

Abstract:

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 $\le 3$ occurrences of a letter are indicated; it is shown that a finitely generated one-relator group $G$ whose reduced relator $R$ is of the form $R \equiv a T_{0} a T_{1} \dots a T_{n-1}$, where the words $T_{i}$ are distinct and have no occurrences of the letter $a^{\pm 1}$, is not hyperbolic if and only if one has in the free group that (1) $n=2$ and $T_{0} T_{1}^{-1}$ is a proper power; (2) $n = 3$ and for some $i$ it is true (with subscripts $\operatorname {mod} 3$) that $T_{i} T_{i+1}^{-1} T_{i} T_{i+2}^{-1} = 1$; (3) $n = 4$ and for some $i$ it is true (with subscripts $\operatorname {mod} 4$) that $T_{i} T_{i+1}^{-1} T_{i+2} T_{i+3}^{-1} = 1$.