Small Cancellation Theory - Generalizations

Generalizations

• A version of small cancellation theory for quotient groups of amalgamated free products and HNN extensions was developed in the paper of Sacerdote and Schupp and then in the book of Lyndon and Schupp.
• Ol'shanskii developed a "stratified" version of small cancellation theory where the set of relators is filtered as an ascending union of stata (each stratum satisfying a small cancellation condition) and for a relator r from some statum and a relator s from a higher stratum their overlap is required to be small with respect to |s| but is allowed to have a large with respect to |r|. This theory allowed Ol'shanskii to construct various "moster" groups including the Tarski monster and to give a new proof that free Burnside groups of large odd exponent are infinite.
• Ol'shanskii and Delzant later on developed versions of small cancellation theory for quotients of word-hyperbolic groups.
• McCammond provided a higher-dimensional version of small cancellation theory.
• McCammond and Wise pushed substantially further the basic results of the standard small cancellation theory (such as Greendlinger's lemma) regarding the geometry of van Kampen diagrams over small cancellation presentations.
• 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. See also for more details on small cancellation theory with respect to a graph.
• Osin gave a version of small cancellation theory for quotiens of relatively hyperbolic groups and used it to obtain a relatively hyperbolic generalization of Thurston's hyperbolic Dehn surgery theorem.