Quotient Tests and Gröbner Bases

Given a finitely presented group G = 〈X;R〉, the word problem asks to decide effectively whether a given word in the free group w ∈ F (X) represents the identity element in G or not. Another important computational problem is to check effectively whether G is finite or not. For the ”No” parts of these problems, we can use quotient tests. For instance, for the ”No” part of the word problem, we can devise a procedure which proves w 6=G 1 or returns "Dont’t know" if we have an effectively computable group homomorphism φ : G −→ H and if the ”No” part of the word problem is decidable in H. To find a suitable group homomorphism φ : G −→ H, we use universal linear representations of G. Special cases of such representations have been used in [12] and [2] for other purposes. Here the universal linear representation % : G −→ SL(n,QR) is constructed as follows. We map each generator xk of G to a matrix of indeterminates (a (k) i,j ). In the polynomial ring P having these indeterminates, we form the ideal IR generated by the entries of the images of the relators in R and by the polynomials det(a (k) i,j )− 1. Letting QR = R/IR, we obtain a well-defined linear representation % : G −→ SL(n,QR) which maps xk to the residue class of (a i,j ).