The conjecture on p.467 of the paper has been proved :
Robert Bieri and J.R.J.Groves, The geometry of the set of characters induced by valuations, J. reine u. angew. Math. 347 (1984) 168-195. MR 86c:14001.
Cf. also
Robert Bieri and J.R.J.Groves, A rigidity principle for the set of all characters induced by valuations, T.A.M.S. 294 (1986) 425-434. MR 87i:16015.
The results of the paper itself have been used by group theorists, e.g.,
J. E. Roseblade, Group rings of polycyclic groups J. Pure Appl. Algebra 3 (1973) 307-328. MR 48#11269.
Another paper extending the ideas of my paper is
Daniel R. Farkas, The Diophantine nature of some constructions at infinity, Geometry of group representations (Boulder, CO, 1987), 125-129, Contemp. Math., 74. MR 89m:20008.
See also the book
Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, v.97, AMS, 2002. viii+152 pp. MR2003i:13037
ERRATA:
The second half of section 3 concerns certain families of subgroups G of Zn which arise as kernels of homomorphisms to ordered groups. Clearly, every such subgroup is pure, equivalently (since Zn is torsion-free), satisfies nx ∈ G ⇒ x ∈ G for positive integers n. With that case in mind, I unfortunately made some general assertions about families of subgroups of Zn that are not true without a purity hypothesis. In particular, the statement on p.462, end of paragraph slightly above the center, "for T > S we have (p1(T), ..., pn(T)) < (p1(S), ..., pn(S)) under lexicographic ordering" is false (e.g., let n = 1, S = {6 Z}, T = {2 Z, 3 Z}). Likewise, in the last sentence of that section, my assertion that Q consists of "nontrivial" subgroups, which as defined in that paper must have infinite index in Zn, does not follow merely from the preceding assertion that Q < {Zn}.
There are several ways these problems can be fixed. The simplest is to define a "subgroup system" to be a finite nonempty family of pure subgroups of Zn. Then the above incorrect assertions become correct. The one modification this entails is in the description of S ∨ T, which will then consist, not of the sums G + H with G ∈ S, H ∈ T, but of the division-closures of those sums in Zn.
A similar solution is to replace the concept of a "subgroup system" by that of a "subspace system", meaning a finite family of vector subspaces of Qn. The general statements about subgroup systems in section 3 become correct for these subspace systems, but a little more work is needed in connecting these results with our hypotheses and conclusions: We must note how to obtain from our homomorphisms on Zn subspaces of Qn, and then having concluded that that the stabilizer of I in GL(n, Z) stabilizes a nontrivial subspace system S, we must note that it therefore stabilizes a nontrivial finite family of subgroups of Zn, namely {G ∩ Zn | G ∈ S}.
As a third alternative, one could keep the original definition of a "subgroup system" (though it is a finer tool than required), and modify our arguments. Despite the incorrectness of the proof using lexicographic order, it is true that the lattice of all subgroup systems of Zn has ascending chain condition; this is a case of a general lemma proved at the end of this errata-page. And the fact, needed at the end of section 3, that Q consists of subgroups of infinite index in Zn can be seen by noting that if it did not, then on dropping all subgroups of finite index, one would have a subgroup system majorizing Q under our partial ordering and still compatible with V.
I am indebted to Greg Marks for pointing out the errors corrected above, and a couple of those noted in the following list of minor corrections:
P.461, last line of first full paragraph: Xα should be xα.
P.463, top line: After "other elements" add "t", and at the end of the line, change "relation" to "relations".
P.464, first sentence of proof of Theorem 2: In (2), &supe should be &sube, and on the next line, after "finally", add "(4)".
P.465, line after first display: In the first inequality, both
occurrences of max should be max(2)
and inversely in the second inequality, both occurrences
of max(2) should be max.
(To see the former inequality, consider two cases:
If |cα xα| is
maximized by the same α that
maximizes |xα|, then all other
terms |cα xα|
involve an |xα| term that
is ≤ max(2)(|xα|), and
the result clearly holds.
In the contrary case, the maximum value
of |cα xα| is
itself ≤
M max(2)(|xα|), hence
a fortiori so is the max(2) value.
The second inequality is straightforward.)
Throughout the paper, I often wrote "designate" where I would now use "denote".
LEMMA REFERRED TO UNDER "ERRATA" ABOVE:
The fact that the set of subgroup systems Zn has ACC is a special case of the following lemma, probably known to people in the field of partially ordered sets. I would be grateful to anyone who could give me a reference. (Note that the antichains of the lemma correspond to the irredundant subgroup systems of the paper.)
Lemma. Let P be any partially ordered set with ACC, and let FA(P) be the partially ordered set of finite antichains in P (finite subsets with no comparable elements), ordered by making T ≥ S if every member of T is ≥ some member of S. Then FA(P) also has ACC.
Proof. Let S0 ≤ S1 ≤ ... in FA(P); it will clearly suffice to show that ∪i Si is finite. Given m ≥ 0, sm ∈Sm and t ∈∪i Si, let us write (m, sm) → t if for some n ≥ m there exists a finite increasing sequence sm ≤ sm+1 ≤ ... ≤ sn = t with si ∈Si for all i. By the hypothesis S0 ≤ S1 ≤ ... and the description of our partial ordering on antichains, we see that for every t ∈∪i Si there is some s0 ∈S0 such that (0, s0) → t. So to show ∪i Si finite, it will suffice to show that for every s0 ∈S0, {t | (0, s0) → t} is finite. We shall in fact show that for every m ≥ 0 and sm ∈ Sm, {t | (m, sm) → t} is finite.
Assuming the contrary, let us choose m ≥ 0 and sm ∈Sm with {t | (m, sm) → t} infinite so as to maximize the element sm ∈P. The infinite set {t | (m, sm) → t} must contain some element ≠ sm, so we may choose n > m and u ∈Sn distinct from sm such that (m, sm) → u. In particular u > sm, so as Sn is an antichain, Sn cannot also contain sm, hence all elements sn ∈Sn that are ≥ sm are strictly greater than it. Hence by the maximality assumption on sm, each of those elements sn ∈Sn has the property that {t | (n, sn) → t} is finite; hence the union of those sets is finite.
But note that any sequence sm ≤ sm+1 ≤ ... ≤ sp with si ∈Si for all i must have the property that all its terms si with i ≥ n (if any) lie in the abovementioned union; while of course all the terms with i < n lie in the finite set Sm ∪ ... ∪ Sn-1. Hence all terms of such sequences, i.e., all elements of {t | (m, sm) → t}, lie in one of these two finite sets, contradicting the assumption that {t | (m, sm) → t} is infinite. \qed
Afterthought: If we associate to every finite antichain S in P the dual ideal (subset of P closed under passing to larger elements) that it generates, we get a bijection between the set of finite antichains and the set of finitely generated dual ideals of P. The ordering described above then corresponds to the opposite of the inclusion ordering on dual ideals. Hence, reversing all orderings, the above lemma translates to say that if P has DCC, then the set of its finitely generated ideals, partially ordered by inclusion, also has DCC. It seems most likely to be known in something like this form to people who study ordered sets.