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Communicated by M. Sapir Dedicated to Stuart Margolis, on the occasion of his 60th birthday.It is well known that under mild conditions on a connected topological space X , connected covers of X may be classified via conjugacy classes of subgroups of the fundamental group of X . In this paper, we extend these results to the study of immersions into two-dimensional CW -complexes. An immersion f : D → C between CW -complexes is a cellular map such that each point y ∈ D has a neighborhood U that is mapped homeomorphically onto f (U ) by f . In order to classify immersions into a two-dimensional CW -complex C, we need to replace the fundamental group of 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. C) by iteratively attaching cells of dimension n to the (n − 1)-skeleton C n−1 of C for n ≥ 1. We refer the reader to Hatcher's text [1], for the precise definition and basic properties of CW -complexes. In particular a continuous map between CWcomplexes is homotopic to a cellular map [1, Theorem 4.8], that is a continuous function that maps cells to cells of the same or lower dimension, so we will regard maps between CW -complexes as cellular maps. A subcomplex of a CW -complex is a closed subspace that is a union of cells.An immersion of a CW -complex D into a CW -complex C is a cellular map f : D → C such that each point y ∈ D has a neighborhood U which is mapped homeomorphically onto f (U ) by f . So f maps n-cells to n-cells. Thus if C is an n-dimensional CW -complex, then D is an m-dimensional CW -complex with m ≤ n. Every subcomplex of an n-dimensional CW -complex C immerses into C. Every covering space of a CW -complex C has a CW -complex structure, and every covering map is in particular an immersion.We classify connected immersions into a two-dimensional CW -complex C via conjugacy classes of closed inverse submonoids of a certain inverse monoid associated with C. The closed inverse submonoids of this inverse monoid enable us to keep track of the 1-cells and 2-cells of C that lift under the immersion, in much the same way as the subgroups of the fundamental group of C enable us to encode coverings of C. We provide an iterative process for constructing the immersion associated with a closed inverse submonoid of this inverse monoid. In many cases this iterative procedure provides an algorithm for constructing the immersion, in particular if the closed inverse submonoid is finitely generated and C has finitely many 2-cells.Section 2 of the paper outlines basic material on presentations of inverse monoids that we will need to build an inverse monoid associated with a 2-complex C. Section 3 describes an iterative procedure for constructing closed inverse submonoids of an inverse monoid from generators for the submonoid. Section 4 describes connected immersions between 2-complexes in terms of a labeling of 1-cells and 2-cells. The main results of the paper linking immersions over a ...

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych C * -algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the C * -algebra.In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Hanoi towers group, the Basilica group, Gupta-Sidki groups, GGS-groups, multiedge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk-Erschler group and Fabrykowski-Gupta group) and self-replicating spinal automaton groups.

In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups.The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies.Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic 0. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.