Groups arising as direct limits of a stationary system of primitive matrices over the integers

Question

I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ integral matrix acting on $\mathbf{Z}^k$ in the usual way and $\displaystyle\lim_{\longrightarrow}$ is the direct limit (colimit) of abelian groups. In the literature these are sometimes referred to as stationary dimension groups.

Such groups will have rank bounded by $k$ and appear as embedded subgroups of $\mathbf{Q}^k$. I'm aware that if we allow the matrices in our directed system to vary, then the direct limit can in general be of 'pathological type' - that is, the direct limit does not necessarily split, up to isomorphism, as a direct sum of groups of the form $\mathbf{Z}[1/n_i]$ for $n_i\geq 1$. The usual proof that these pathological colimits can appear relies on a cardinality argument; there are an uncountable number of isomorphism classes of direct limits of matrices, but only a countable number of finite rank non-pathological type isomorphism classes of groups.

My question is does the stationary case, where $M$ is a fixed matrix and cannot vary, allow for direct limits of pathological type? A counting argument won't work in this setting because we now only have a countable number of stationary direct limits, and this appears to lend support for the conjecture that such pathological direct limits cannot arise.

I'm not an algebraist and so wading through the mountains of literature surrounding the theory of dimension groups is proving difficult. I would appreciate any insight/references.

Answer 1

Generically, the limits of stationary sequences are much much nastier than you anticipate; for example, some strongly indecomposable abelian groups of rank exceeding one can be the underlying group for the direct limit (this usually happens if the matrix is 2 x 2, its Perron eigenvalue is irrational, the determinant and the trace have no common divisors, and the determinant is not plus or minus one). Even if the large eigenvalue, $n$, is an integer, it's still generically true that the natural map $G \to {\bf Z}[1/n]$ doesn't split, and usually there is what you called pathological behaviour---it is generic. For example, see Boyle, Mike; Handelman, David Algebraic shift equivalence and primitive matrices. Trans. Amer. Math. Soc. 336 (1993), no. 1, 121–149, particularly the examples, which exhibit the limits as extensions.

On the other hand, the dimension groups that arise as limits of stationary systems are limited by their (continuous) endomorphism ring, essentially a ring of matrices (and some formal inverses) that commute with the original matrix, over which the limit group is a module of "rank" one (care must be taken with the use of rank here). For example, Positive matrices and dimension groups affiliated to $C^{\ast} $-algebras and topological Markov chains. J. Operator Theory 6 (1981), no. 1, 55–74, and Matrices of positive polynomials. Electron. J. Linear Algebra 19 (2009), 2–89 (the section dealing with the case of zero variables).

Here is a newer reference, which gives a few more stationary examples that are usually far from "non-pathological". Realizing dimension groups, good measures, and Toeplitz factors, Ill J Math 57 (2013*), 1057--1109.

*2013 is fiction; the issue appeared this month (December 2014), but they are far behind in publication.