Undecidable properties of monoids with word problem solvable in linear time. Part II— cross sections and homological and homotopical finiteness conditions

Abstract

Using a particular simulation of single-tape Turing machines by finite string-rewriting systems the first two authors have shown that all linear Markov properties are undecidable for the class of finitely presented monoids with linear-time decidable word problem. Expanding on this construction it is shown here that also many properties that are not known to be linear Markov properties are undecidable for this class of monoids. These properties include the existence of context-free or regular cross-sections, the existence of finite convergent presentations, the property of being automatic, and the homological and homotopical finiteness properties left- and right-FPn(n⩾3), FHT, and FDT.