text: Novikov–Boone theorem The negative solution to the word problem for groups states that there is a finite presentation    for which there is no algorithm which, given two words u, v, decides whether u and v describe the same element in the group.  This was shown by Pyotr Novikov in 1955 and a different proof was obtained by William Boone in 1958.
  Constructions
  Suppose G has presentation  and H has presentation  with S and T being disjoint. Then
  the free product  has presentation  and
  the direct product  has presentation , where [S, T] means that every element from S commutes with every element from T (cf. commutator).
  Deficiency
  The deficiency of a finite presentation  is just  and the deficiency of a finitely presented group G, denoted def(G), is the maximum of the deficiency over all presentations of G.  The deficiency of a finite group is non-positive.  The Schur multiplicator of a finite group G can be generated by −def(G) generators, and G is efficient if this number is required.
  Geometric group theory
  A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
  Further, some properties of this graph (the coarse geometry) are intrinsic, meaning independent of choice of generators.