Inverse monoids and immersions of 2-complexes

It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathcal C$, we need to replace the fundamental group of $\mathcal C$ by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex

The uniform Roe algebra of an inverse semigroup

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Sch\"{u}tzenberger graphs. This graph, which we denote by $\Lambda_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(\Lambda_S)$. This yields a chacterization of when $\mathcal{R}_S = C^*_u(\Lambda_S)$, which generalizes finite generation of $S$. We have termed this by finite labeability (FL), since it holds when the $\Lambda_S$ can be labeled in a finitary manner. The graph $\Lambda_S$, and the FL condition above, also allow to analyze large scale properties of $\Lambda_S$ and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of $S$ (a notion generalizing Day's definition of amenability of a semigroup, cf., [5]) is a quasi-isometric invariant of $\Lambda_S$. Moreover, we characterize property A of $\Lambda_S$ (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view.Comment: Accepted version for publication; minor change

Algorithmic properties of inverse monoids with hyperbolic and tree-like Sch\"utzenberger graphs

We prove that the class of finitely presented inverse monoids whose Sch\"utzenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Sch\"utzenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Sch\"utzenberger graphs and an unsolvable word problem

Amenability and coarse geometry of (inverse) semigroups and C*-algebras

Mención Internacional en el título de doctorThis Doctoral Thesis has been supported by the Severo-Ochoa grants SEV-2015- 0554 and BES-2016-077968, from the Spanish Ministry of Economy and Competition (MINECO), via the Instituto de Ciencias Matemáticas (ICMAT) of the Centro Superior de Investigaciones Científicas (CSIC). Additionally, the author has been partially supported by the Ministry of Economy and Competitiveness of Spain under Research Project MTM2017-84098-P.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Enrique Pardo Espino.- Secretario: Luis Alberto Ibort Latre.- Vocal: Nadia S. Larse