10 research outputs found
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
String rewriting for Double Coset Systems
In this paper we show how string rewriting methods can be applied to give a
new method of computing double cosets. Previous methods for double cosets were
enumerative and thus restricted to finite examples. Our rewriting methods do
not suffer this restriction and we present some examples of infinite double
coset systems which can now easily be solved using our approach. Even when both
enumerative and rewriting techniques are present, our rewriting methods will be
competitive because they i) do not require the preliminary calculation of
cosets; and ii) as with single coset problems, there are many examples for
which rewriting is more effective than enumeration.
Automata provide the means for identifying expressions for normal forms in
infinite situations and we show how they may be constructed in this setting.
Further, related results on logged string rewriting for monoid presentations
are exploited to show how witnesses for the computations can be provided and
how information about the subgroups and the relations between them can be
extracted. Finally, we discuss how the double coset problem is a special case
of the problem of computing induced actions of categories which demonstrates
that our rewriting methods are applicable to a much wider class of problems
than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio
Polynomial-time proofs that groups are hyperbolic
Funding: UK EPSRC grant number EP/I03582X/1.It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings (1971), to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure that analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and when successful they are many orders of magnitude faster than KBMAG, the only comparable publicly available software.PostprintPeer reviewe