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Rewriting techniques in theoretical physics
Evelyne Contejean, Antoine Coste and Benjamin Monate
This paper presents a general method for studying some quotients of
the special linear group SL2 over the integers, which are of
fundamental interest in the field of statistical physics. Our method
automatically helps in validating some conjectures due to physicists,
such as conjectures stating that a set of equations completely
describes a finite given quotient of SL2. In a first step, we show
that in the cases we are interested in, the usual presentation of
finitely generated groups with some constant generators and a binary
concatenation can be turned into an equivalent one with unary
generators. In a second step, when the completion of the transformed
set of equations terminates, we show how to compute directly the
associated normal forms automaton. According to the presence of
loops, we are able to decide the finiteness of the quotient, and to
compute its cardinality. When the quotient is infinite, the automaton
gives some hints on what kind of equations are needed in order to
insure the finiteness of the quotient.