Honolulu, HI 96822
Title: Degrees of the isomorphism types of structures
The Turing degree spectrum of a countable structure $A$ is the set of all Turing degrees of the isomorphic copies of $A$.
Knight proved that the degree spectrum of a structure is closed upwards, unless the structure is automorphically trivial, in which case the degree spectrum consists of a single Turing degree.
Hirschfeldt, Khoussainov, Shore, and Slinko established that for every automorphically nontrivial structure $A$, there is a symmetric irreflexive graph, a partial order, a lattice, a ring, an integral domain of arbitrary characteristic, a commutative semigroup, or a $2 $-step nilpotent group the degree spectrum of which coincides with the degree spectrum of $A$.
Jockusch and Richter defined the degree of the isomorphism type of a structure $A$ to be the least Turing degree in the degree spectrum of $A$.
Such degrees may not exist.
For example, Richter showed that linear orders and trees without computable isomorphic copies do not have degrees of their isomorphism types.
A.N. Khisamiev established the same result for abelian $p$-groups.
Richter also introduced a general combination method for building structures the isomorphism types of which have arbitrary Turing degrees.
As a corollary, she showed that there is an abelian torsion group with an arbitrary degree of its isomorphism type.
We will show how Richter’s combination method can be extended.
We will present some recent results on the degrees of the isomorphism types of structures of interest in algebra, geometry, and low-dimensional topology.
These results are obtained in collaboration with different groups of researchers.