The word problem for some uncountable groups given by countable words

We investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application we give a short proof that these homology groups contain the direct sum of 2^{\aleph_0} copies of \mathbb{Q}. Finally, we show that the fundamental group of Griffith's space contains \mathbb{Q}.Comment: 24 pages, 7 figure

The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter B\"uchi automaton and a 2-tape B\"uchi automaton such that the \omega-language of the first and the infinitary rational relation of the second in one model of ZFC are \pi_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by B\"uchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author

On structures in hypergraphs of models of a theory

We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types of models of a theory, are given

Borel Conjecture and Dual Borel Conjecture

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.Comment: 47 pages, revised version 2013 (some typos removed, some points elaborated. Dedication added.

On the number of Mather measures of Lagrangian systems

In 1996, Ricardo Ricardo Ma\~n\'e discovered that Mather measures are in fact the minimizers of a "universal" infinite dimensional linear programming problem. This fundamental result has many applications, one of the most important is to the estimates of the generic number of Mather measures. Ma\~n\'e obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able with Gonzalo Contreras to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest