Connective Bieberbach Groups

Ellen L Weld, Purdue University

Abstract

Connectivity is a lifting property of C ∗ -algebras originally isolated by Dadarlat and Pennig in 2017. In their remarkable paper, they showed that connectivity completely characterizes the separable nuclear C ∗ -algebras whose E-theory may be unsuspended; that is, A is connective if and only if E(A, B) := [[SA, SB ⊗ K]] ∼= [[A, B ⊗ K]] for all separable C ∗ -algebrasB. Connectivity offers a wealth of permanence properties including passing to C ∗ -subalgebras and split extensions — properties not obvious when viewed from a purely E-theoretical perspective. By the contributions of Dadarlat and Pennig and Gabe, connectivity of a C ∗-algebra A was also shown to be equivalent to its primitive spectrum containing no non-empty compact open subsets. Although investigating the primitive spectrum is still a challenge, this does provide a testable criterion for connectivity. Bieberbach groups, of independent interest in physics and chemistry, are exactly the fundamental groups of flat compact Riemannian manifolds. Abstractly, these are torsion free groups fitting into an exact sequence of the form 1 → Z n→ G → D → 1 where :D: < ∞. The unitary dual (equivalently the primitive spectrum of the associated group C ∗-algebra) of Bieberbach groups benefit from Mackey’s machine which allows us to build the unitary dual from “smaller” representations of subgroups. This means that, by careful investigation, we may determine under what conditions the topology of the unitary dual contains non-empty compact open subsets. The primary result of this document is Theorem 6.1 which shows that if a Bieberbach group has finite abelianization, then its unitary dual contains a compact open subset and thus is not connective. The proof of this theorem is a generalization of a result in Dadarlat and Pennig’s paper [1] which shows that the Hantzsche-Wendt group is not connective. Combining this result with other work on Bieberbach groups, we determine that a Bieberbach group G is connective if and only if no non-trivial subgroup of Ghas finite abelianization.

Degree

Ph.D.

Advisors

Dadarlat, Purdue University.

Subject Area

Mathematics

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