Division in group rings of surface groups

Abstract

We prove a division algorithm for group rings of high genus surface groups and use it to show that some 2-complexes with surface fundamental groups are standard. We also give an application of division to cohomological dimension of 2-relator groups acting on H n .1 To take a concrete example, in the free group F2 on the letters g and h, any linear combination of g − 1 and h − 1 has diameter ≥ 1, so it is not possible to divide g − 1 by h − 1 with remainder of zero diameter.2 In the commutative case QF1 = Q[t, t −1 ] any pair of elements x and y satisfy the obvious relation xy−yx = 0, and the algorithm becomes the usual long division for Laurent polynomials.