A Criterion for Hyperbolicity

The usual deenition of hyperbolicity of a group G demands that all geodesic triangles in the Cayley graph of G should be thin. Using the theorem that a subquadratic isoperimetric inequality implies a linear one, we show that it is in fact only necessary for all triangles from a given combing to be thin, thus giving a new criterion for hyperbolicity of nitely presented groups. Given a group G the Cayley graph ? S (G) of G with respect to a generating set S of G is the graph whose vertex set is G and whose edge set is f(g; gs)jg 2 G; s 2 Sg. Given a path p in ? S (G) we write l(p) for the number of edges in p. If p originates at the identity of G then we write p for the group element at the terminus of p (i.e. p is the group element represented by the word p in S). where g 1 , g 2 , and g 3 are elements of G called the vertices of the triangle and ij is a path in the Cayley graph of G from g i to g j (called a side of the triangle). If the sides are geodesic paths, the triangle is said to be geodesic. For a triangle as above, we denote by @ the loop 12 23 31 , called the boundary of and we write (() for l(@), the perimeter of. The following deenition is based on the familiar geodesic case.