| Hyperbolic groups were introduced about 25 years ago. They include finite groups, free groups, free products, free factors, direct factors, taking subgroups or supergroups of finite index, and so on. We would like to study an Ldelta-group which is a generalization of a hyperbolic group.;We observe that the hyperbolic groups may not be closed under the direct product. For instance, ZxZ is not hyperbolic while Z is hyperbolic. We prove that a direct product of L delta-groups is an Ldelta-group. Also, we show the following: finite-by-Ldelta-groups are Ldelta-groups, virtually L delta-groups are Ldelta-groups, and abelian groups are Ldelta-groups.;We calculate the Dehn function or an isoperimetric function of an Ldelta-group. It is known that a group is word problem solvable if and only if its Dehn function (or an isoperimetric function) is recursive. Hyperbolic groups are characterized as the groups with linear Dehn functions. We prove that Ldelta-groups have sub-cubic Dehn functions so that Ldelta-groups are word problem solvable. |
