Examples of applications of small cancellation theory include: • Solution of the
conjugacy problem for groups of
alternating knots (see and Chapter V, Theorem 8.5 in ), via showing that for such knots augmented knot groups admit C(4)–T(4) presentations. • Finitely presented
C′(1/6) small cancellation groups are basic examples of
word-hyperbolic groups. One of the equivalent characterizations of word-hyperbolic groups is as those admitting finite presentations where Dehn's algorithm solves the
word problem. • Finitely presented groups given by finite C(4)–T(4) presentations where every piece has length one are basic examples of
CAT(0) groups: for such a presentation the
universal cover of the
presentation complex is a
CAT(0) square complex. • Early applications of small cancellation theory involve obtaining various embeddability results. Examples include a 1974 paper of Sacerdote and Schupp with a proof that every one-relator group with at least three generators is
SQ-universal and a 1976 paper of Schupp with a proof that every countable group can be embedded into a
simple group generated by an element of order two and an element of order three. • The so-called
Rips construction, due to
Eliyahu Rips, provides a rich source of counter-examples regarding various
subgroup properties of
word-hyperbolic groups: Given an arbitrary finitely presented group
Q, the construction produces a
short exact sequence 1\to K\to G\to Q\to 1 where
K is two-generated and where
G is torsion-free and given by a finite
C′(1/6)–presentation (and thus
G is word-hyperbolic). The construction yields proofs of unsolvability of several algorithmic problems for
word-hyperbolic groups, including the subgroup membership problem, the generation problem and the
rank problem. Also, with a few exceptions, the group
K in the Rips construction is not
finitely presentable. This implies that there exist word-hyperbolic groups that are not
coherent that is which contain subgroups that are finitely generated but not finitely presentable. • Small cancellation methods (for infinite presentations) were used by Ol'shanskii •
Bowditch used infinite small cancellation presentations to prove that there exist continuumly many
quasi-isometry types of two-generator groups. • Thomas and Velickovic used small cancellation theory to construct a
finitely generated group with two non-homeomorphic asymptotic cones, thus answering a question of
Gromov. • McCammond and Wise showed how to overcome difficulties posed by the Rips construction and produce large classes of small cancellation groups that are
coherent (that is where all finitely generated subgroups are finitely presented) and, moreover, locally quasiconvex (that is where all finitely generated subgroups are quasiconvex). • Small cancellation methods play a key role in the study of various models of "generic" or
"random" finitely presented groups (see ). In particular, for a fixed number
m ≥ 2 of generators and a fixed number
t ≥ 1 of defining relations and for any
λ εn (where
ε ≥ 0 is the fixed
density parameter in Gromov's density model of "random" groups, and where n\to\infty is the length of the defining relations), then an
ε-random group satisfies the
C′(1/6) condition provided
ε < 1/12. •
Gromov used a version of small cancellation theory with respect to a graph to prove the existence of a
finitely presented group that "contains" (in the appropriate sense) an infinite sequence of
expanders and therefore does not admit a uniform embedding into a
Hilbert space. This result provides a direction (the only one available so far) for looking for counter-examples to the
Novikov conjecture. • Osin used a generalization of small cancellation theory to obtain an analog of
Thurston's hyperbolic Dehn surgery theorem for
relatively hyperbolic groups. ==Generalizations==