On Dehn functions of infinite presentations of groups

Rostislav I. Grigorchuk, Sergei V. Ivanov

Research output: Contribution to journalArticlepeer-review


We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups. In particular, we prove that the (standard) Dehn function of a 2-group Γ of intermediate growth, defined by a system of defining relators due to Lysenok, is bounded from above by C 1 x 2 log 2 x, where C 1 > 1 is a constant. We also show that the (standard) Dehn function of a free m-generator Burnside group B(m, n) of exponent n ≥ 248, where n is either odd or divisible by 2 9, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x 19/12.

Original languageEnglish (US)
Pages (from-to)1841-1874
Number of pages34
JournalGeometric and Functional Analysis
Issue number6
StatePublished - Mar 2009


  • Burnside groups
  • Dehn functions
  • Presentation of groups
  • Torsion groups
  • Van Kampen diagrams
  • Word problem

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology


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