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We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and…
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…
The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative $C^*$-algebras. We study the relationship between the geometry of an inverse…
We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an…
We define and study the notion of a crossed module over an inverse semigroup and the corresponding $4$-term exact sequences, called crossed module extensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we show that equivalence…
The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Jarek is that commutative inverse monoids become semilattices of abelian groups. It has…
We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to…
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…
We present some homological properties of a relation $\beta$ on ordered groupoids that generalises the minimum group congruence for inverse semigroups. When $\beta$ is a transitive relation on an ordered groupoid $G$, the quotient $G /…
We introduce a formalism of infinite, linearly ordered products in general groups. Using this, we define infinite compositions in certain groups of formal power series such as transseries. We show that such groups can sometimes be…
We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one…
Here we prove that for dilatation structures linearity (see arXiv:0705.1440v1) is equivalent to a statement about the inverse semigroup generated by the family of dilatations of the space. The result is new for Carnot groups and the proof…
We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$…
We fix a path model for the space of filters of the inverse semigroup $\mathcal{S}_\Lambda$ associated to a left cancellative small category $\Lambda$. Then, we compute its tight groupoid, thus giving a representation of its $C^*$-algebra…
In this paper subcentral (resp., central) idempotent series and composition subcentral (resp., central) idempotent series in an inverse semigroup are introduced and investigated. It is shown that if $S=EG$ is a factorizable inverse monoids…
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call,…
The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which…
In this paper we give complete analytic invariants for germs of holomorphic foliations in $(\mathbb{C}^2,0)$ that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse…
Various characterizations are offered of injectivity of the canonical fundamental group homomorphism for a certain class of inverse limit spaces. One application characterizes the existence of a kind of generalized universal cover.
For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced…