W. A. Bogley and A. J. Sieradski, Weighted combinatorial group theory and wild
metric complexes, 11 August, 1997
Abstract
In this paper, we develop the low dimensional homotopy theory required for
weighted combinatorial group theory. In Omega-groups, an earlier paper by Sieradski, the usual concepts of
generators and relators of group presentations are extended to weighted
generators and weighted relators for weighted group presentations. This
extension parallels the passage from finite sets to order types, i.e. closed
nowhere dense sets in the closed unit interval. In the weighted environment,
products of all order-type are permitted, provided that the entries of the
product have weights that limit at zero as their depth of occurrence in the
order-type increases without bound. Here, we develop weighted analogs of the
usual correspondence via fundamental groups between free groups and
1-dimensional CW cell complexes and between group presentations and
2-dimensional CW cell complexes. The results are a correspondence between free omega-groups and wild metric 1-complexes in which the 1-cells can
limit on 0-cells and a correspondence between weighted group
presentations and wild metric 2-complexes in which the 1-cells and
2-cells can limit on the 0-cells.