Subgroups of direct products of two limit groups

If S is a subgroup of a direct product of two limit groups, and S is of type FP(2) over the rationals, then S has a subgroup of finite index that is a direct product of at most two limit groups.Comment: 18 pages, no figure

The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.Comment: 15 page

Symmetrization of Rational Maps: Arithmetic Properties and Families of Latt\`es Maps of Pk\mathbb P^k

In this paper we study properties of endomorphisms of $\mathbb P^k$ using a symmetric product construction $(\mathbb P^1)^k/\mathfrak{S}_k \cong \mathbb P^k$. Symmetric products have been used to produce examples of endomorphisms of $\mathbb P^k$ with certain characteristics, $k\geq2$. In the present note, we discuss the use of these maps to enlighten arithmetic phenomena and stability phenomena in parameter spaces. In particular, we study notions of uniform boundedness of rational preperiodic points via good reduction information, $k$-deep postcritically finite maps, and characterize families of Latt\`es maps.Comment: Added more background and references; repaired a small gap in Lemma 3.1; reordered some statements in Propositions 1.1 and 1.2; 26 page