Finitely generated abelian groups of units

In 1960 Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper we address Fuchs' question for {\it finitely generated abelian} groups and we consider the problem of characterizing those groups which arise in some fixed classes of rings $\mathcal C$, namely the integral domains, the torsion free rings and the reduced rings. To determine the realizable groups we have to establish what finite abelian groups $T$ (up to isomorphism) occur as torsion subgroup of $A^*$ when $A$ varies in $\mathcal C$, and on the other hand, we have to determine what are the possible values of the rank of $A^*$ when $(A^*)_{tors}\cong T$. Most of the paper is devoted to the study of the class of torsion-free rings, which needs a substantially deeper study.Comment: 28 page

Entropy in Dimension One

This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite self-map of the unit interval if and only if exp(h) is an algebraic integer that is at least as large as the absolute value of any of the conjugates of exp(h); that is, if exp(h) is a weak Perron number. The postcritically finite map may be chosen to be a polynomial all of whose critical points are in the interval (0,1). This paper also proves that the weak Perron numbers are precisely the numbers that arise as exp(h), where h is the topological entropy associated to ergodic train track representatives of outer automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before his death, and was uploaded by Dylan Thurston. A version including endnotes by John Milnor will appear in the proceedings of the Banff conference on Frontiers in Complex Dynamic

Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's pp-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements