Small Cancellation Theory - Applications

Applications

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(T)–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 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 to construct various "monster" groups, including the Tarski monster and also to give a proof that free Burnside groups of large odd exponent are infinite (a similar result was originally proved by Adian and Novikov in 1968 using more combinatorial methods). Some other "monster" groups constructed by Ol'shanskii using this methods include: an infinite simple Noetherian group; an infinite group in which every proper subgroup has prime order and any two subgroups of the same order are conjugate; a nonamenable group where every proper subgroup is cyclic; and others.
• 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 < 1 a random m-generator t-relator group satisfies the C'(λ) small cancellation condition. Even if the number of defining relations t is not fixed but grows as (2m−1)εn (where ε ≥ 0 is the fixed density parameter in Gromov's density model of "random" groups, and where 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.