An alternative proof that the Fibonacci group F(2,9) is infinite

This note contains a report of a proof by computer that the Fibonacci group F(2,9) is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that the group generators have infinite order, which of course implies that the group itself is infinite.Comment: LaTex, 3 pages, no figures. To appear in Experimental Mathematic

Solving the word problem in real time

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems

Finitely generated soluble groups and their subgroups

We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.Comment: 16 page

On real-time word problems

It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the Baumslag–Solitar groups B(1,r) have the 5-tape real-time word problem for all r != 0

Sofic groups: graph products and graphs of groups

We prove that graph products of sofic groups are sofic, as are graphs of groups for which vertex groups are sofic and edge groups are amenable