The symmetries of outer space

Published versio

The triviality problem for profinite completions

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion.This is the accepted manuscript. The final version is available from Springer at http://dx.doi.org/10.1007/s00222-015-0578-

La construcción simbólica de una capital. Planeamiento, imagen turística y desarrollo urbano en Barcelona a principios de siglo XX

La producción iconográfica de las ciudades constituye una documentación imprescindible para los estudios de historia urbana pero, a la vez, son elementos con un gran valor interpretativo de los procesos de control urbanístico y, por tanto, de las estrategias de intervención asociadas a intereses de tipo económico, social y cultural. Estas producciones, en forma de vistas y cartografía urbana, son todavía más apreciables cuando son el resultado de la correlación entre los momentos álgidos de la planificación urbanística, el desarrollo urbano real y la creación de un relato de apropiación de la ciudad. La representación iconográfica posee una capacidad comunicativa más efectiva que la derivada del planeamiento, por lo que elabora una construcción simbólica de una realidad urbana con fuerte persistencia en el imaginario cultural. El caso de la ciudad de Barcelona y la cartografía turística de principios de siglo XX es un ejemplo intenso e interesante de esta relación

Undecidability and the developability of permutoids and rigid pseudogroups

A $\textit{permutoid}$ is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.Bridson and Wilton thank the EPSRC for its financial support. Bridson’s work is also supported by a Wolfson Research Merit Award from the Royal Society

Algorithms determining finite simple images of finitely presented groups

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, for any fixed untwisted Lie type X there is an algorithm that determines whether or not any given finitely presented group has simple images of the form X(q) for infinitely many q, and if there are finitely many, the algorithm determines them

On the Commutative Equivalence of Context-Free Languages

The problem of the commutative equivalence of context-free and regular languages is studied. In particular conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated

Minsky machines and algorithmic problems

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.Comment: 19 page

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Continuous selections of Lipschitz extensions in metric spaces

This paper deals with the study of parameter dependence of extensions of Lipschitz mappings from the point of view of continuity. We show that if assuming appropriate curvature bounds for the spaces, the multivalued extension operators that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are lower semi-continuous and admit continuous selections. Moreover, we prove that Lipschitz mappings can be extended continuously even when imposing the condition that the image of the extension belongs to the closure of the convex hull of the image of the original mapping. When the target space is hyperconvex one can obtain in fact nonexpansivity.Dirección General de Enseñanza SuperiorJunta de AndalucíaRomanian Ministry of Educatio