We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$-algebras on one-dimensional solenoids.

Abstract. We define canonical subshift of finite type covers for Williams' onedimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids.

Abstract. We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the standard order of the adjacency matrices associated with the presentation. Introduction.Ordered groups have been useful invariants for the classification of many different categories. A class of ordered groups, dimension groups, was used in the study of C * -algebras to classify AF-algebras ([6]), and Giordano, Herman, Putnam and Skau ([8,9]) defined (simple) dimension groups in terms of dynamical concepts to give complete information about the orbit structure of zero-dimensional minimal dynamical systems. Swanson and Volkmer ([15]) showed that the dimension group of a primitive matrix is a complete invariant for weak equivalence, which is called C * -equivalence by Bratteli, Jørgensen, Kim and Roush ([5]). And Barge, Jacklitch and Vago ([3]) showed that, for a certain class of one-dimensional inverse limit spaces, two spaces are homeomorphic if and only if their associated substitutions are weak equivalent, and that if two inverse limit spaces are homeomorphic and the squares of their connection maps are orientation preserving, then the dimension groups of the adjacency matrices associated with the substitutions are order isomorphic.A recent development ([2, 3, 4, 7, 8, 15]) is the refinement ofȞ 1 (X) as a topological invariant for certain one-dimensional spaces X, by making this group an ordered group. HereȞ 1 (X) is the direct limit of first cohomology groups on graphs approximating the space X. There is a natural order on the first cohomology of a graph (a coset is positive if it contains a nonnegative

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