Computable paradoxical decompositions

Baire Measurable Paradoxical Decompositions via Matchings

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall’s condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theore...

Paradoxical Decompositions, 2-generator Kleinian Groups, and Volumes of Hyperbolic 3-manifolds

The 2-thin part of a hyperbolic manifold, for an arbitrary positive number 2, is defined to consist of all points through which there pass homotopically non-trivial curves of length at most 2. For small enough 2, the 2-thin part is geometrically very simple: it is a disjoint union of standard neighborhoods of closed geodesics and cusps. (Explicit descriptions of these standard neighborhoods are...

Lecture 1 : Paradoxical decompositions of groups and their actions . by Kate Juschenko

Computable choice functions for computable linear orderings

One direction of research in computability theory is the study of the effective content of combinatorial theorems. The last chapter of Rosenstein’s book [2] reports on this type of investigation for linear orderings. Questions raised in that chapter have been the subject of a number of papers, and a summary of results in the area can be found in Downey [1]. In this paper, we consider the proble...

Lowness for Computable and Partial Computable Randomness

Ambos-Spies and Kucera [1, Problem 4.8] asked whether there is a non-computable set which is low for the computably random sets. We show that no such set exists. The same result holds for partial computable randomness. Each tally language that is low for polynomial randomness is on a polynomial time tree of bounded width. This research was done in 2003 but has not been published so far.