Narrow Results

Let Γ=ℤ_m * ℤ_n or ℤ * ℤ_n, and let Γ(h) be the subtree consisting of all reduced words starting with any reduced word h ∈ Γ\{e}. We prove that the C*-algebra generated by and the projection P_h onto the subspace ℓ²(Γ(h)) has a unique nontrivial closed ideal ℐ, ℐ is *-isomorphic to , and the quotient algebra is *-isomorphic to either or depending on the last letter of h. We also prove that is a purely infinite, simple C*-algebra if the last letter of h is a generator of ℤ, and that has a unique nontrivial closed ideal if the last letter of h is a generator of ℤ_n; furthermore, is *-isomorphic to and is again a purely infinite, simple C*-algebra. As consequences, all the C*-algebras above have real rank zero, and is nuclear for any h ≠ e.

We study link-homotopy classes of links in the three sphere using reduced groups endowed with peripheral structures derived from meridian-longitude pairs. Two types of peripheral structures are considered — Milnor’s original version (called “pre-peripheral structures” in Levine’s terminology) and Levine’s refinement (called simply “peripheral structures”). We show here that pre-peripheral structures are not strong enough to classify links up to link-homotopy, and that Levine’s peripheral structures, although strong enough to distinguish those classes not distinguished by pre-peripheral structures, are also in all likelihood not strong enough to distinguish all link-homotopy classes. Following Levine’s classification program, we compare structure-preserving and realizable automorphisms, using an obstruction-theoretic approach suggested by work of Habegger and Lin. We find that these automorphism groups are in general different, so that a more complex program for classification by structured groups is required.