relator groups like Baumslag–Solitar groups. These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite. The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a deficiency zero. For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency. An efficient group is one where the Schur multiplier requires this number of generators. A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers. Such presentations are in some sense nice because they are usually short, but they are difficult to find and to work with because they are ill-suited to standard methods such as coset enumeration. Relation to topology In topology, groups can often be described as finitely presented groups and a fundamental question is to calculate their integral homology . In particular, the second homology plays a special role and this led Heinz Hopf to find an effective method for calculating it. The method in is also known as Hopf's integral homology formula and is identical to Schur's formula for the Schur multiplier of a finite group: where and F is a free group. The same formula also holds when G is a perfect group. The recognition that these formulas were the same led Samuel Eilenberg and Saunders Mac Lane to the creation of cohomology of groups. In general, where the star denotes the algebraic dual group. Moreover, when G is finite, there is an unnatural isomorphism The Hopf formula for has been generalised to higher dimensions. For one approach and references see the paper by Everaert, Gran and Van der Linden listed below. A perfect group is one whose first integral homology vanishes. A superperfect group is one whose first two integral homology groups vanish. The Schur covers of finite perfect groups are superperfect. An acyclic group is a group all of whose reduced integral homology vanishes. Applications The second algebraic K-group K2(R) of a commutative ring R can be identified with the second homology group H2(E(R),