Abstract.
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 C1x2 log2 x, where C1 > 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 29, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x19/12.
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The first named author is partially supported by NSF grants DMS 04-56185, DMS 06-00975 and by the Swiss National Science Foundation. The second named author is supported in part by NSF grant DMS 04-00746
Received: September 2007, Revision: March 2008, Accepted: March 2008
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Grigorchuk, R.I., Ivanov, S.V. On Dehn Functions of Infinite Presentations of Groups. GAFA Geom. funct. anal. 18, 1841–1874 (2009). https://doi.org/10.1007/s00039-009-0712-0
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DOI: https://doi.org/10.1007/s00039-009-0712-0