Speaker: Steffen Lempp
University of Wisconsin
Will give a Logic Seminar entitled:
Some results on the Ziegler degrees
Abstract: In a landmark 1980 monograph, Ziegler studied finitely generated subgroups of existentially closed groups and uncovered a surprising connection of this purely algebraic question and computability. To this end, he defined a new reducibility which he called $*$-reducibility and which I will call Ziegler reducibility, which is a refinement of both Turing and enumeration reducibility. In particular, he showed that if a finitely generated group~$H$ is a subgroup of an existentially closed group~$G$ and the word problem of a finitely generated group~$H_0$ is Ziegler reducible to the word problem of~$H$, then~$H_0$ is (isomorphic to) a subgroup of~$G$ as well. In joint work with Isabella Scott and Josiah Jacobsen-Grocott, I studied the induced degree structure of the Ziegler degrees and was able to show that there is a minimal Ziegler degree, and indeed that any finite distributive lattice is isomorphic to an initial segment of the Ziegler degrees. As a consequence, the $\forall\exists\forall$-theory of the Ziegler degrees (as a partial order) is undecidable.
Date: 08-27-2024
Time: 2:00 pm
Location: 125 Hayes-Healy Bldg