Average-case complexity and decision problems in group theory

Abstract. We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on "generic-case complexity" we show that if a finitely generated group G has the word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem for G is linear time, uniformly with respect to the collection of all length-invariant measures on G. For example, the result applies to all braid groups Bn.

• Type: Journal
• Year: 2002
• Where: CORR
• Authors: Ilya Kapovich, Alexei G. Myasnikov, Paul Schupp, Vladimir Shpilrain
• Keywords: Average-case Complexity, CORR 2002, Education, Finitely Generated Group, Word Problem