On higher order homological finiteness of rewriting systems

Abstract

A sequence of finite rewriting systems R n ( n ⩾ 1 ) with the following properties is presented: (1) R n is not of type FDT, (2) R n is of type FHT n , but not of type FHT n + 1 , (3) R n has word problem solvable in quadratic time. This result not only strengthens the result of Pride and Otto separating the geometric finiteness condition FDT from the finiteness condition FHT ( = FHT 1 ), but it also shows that the higher order homological finiteness conditions FHT n , which were first considered by McGlashan for dimension 2, yield an infinite hierarchy that is independent of the homotopical finiteness condition FDT.