Completely decomposable summands of almost completely decomposable groups

Abstract

Almost completely decomposable groups are torsion-free finite extensions of completely decomposable groups of finite rank. We answer completely and in a constructive fashion the question when an almost completely decomposable group has non-zero completely decomposable direct summands. A new invariant, the rank-width difference of X at τ, given by

$$rw{d_\tau }\left( X \right) = rk\frac{{X\left( \tau \right)}} {{{X^\# }\left( \tau \right)}} - width\frac{{{X^\# }\left[ \tau \right]}} {{X\left( \tau \right) + X\left( \tau \right)}}$$

is the exact rank of a maximal τ-homogeneous (completely decomposable) direct summand of X. We show that the rank-width difference can be effectively computed for an almost completely decomposable group given in “standard description” X = A + f ℕN ^-1 α^l provided that A is regulating in X. We also establish an algorithmic criterion for deciding whether an almost completely decomposable group given in standard description is completely decomposable.

The hospitality and support of the University of the Western Cape during February 1998 is gratefully recognized.