Two fundamental finiteness properties in group theory are those of being finitely generated and of being finitely presented. These two finiteness properties were generalised to higher dimensions by C. T. C. Wall in 1965. A group G is said to be of type Fn if it has an Eilenberg–MacLane complex K(G,1) with finite n-skeleton. (Here K(G,1) is a certain nice topological space with fundamental group G.) It may be shown that a group is finitely generated if and only if it is of type F1 and is finitely presented if and only if it is of type F2. Related to this is a certain homological finiteness property called FPn. This property is defined for a monoid S in terms of the existence of certain resolutions of free left ZS-modules. The property FPn was introduced for groups by Bieri in (1976). In monoid and semigroup theory the property FPn arises naturally in the study of string rewriting systems. The connection between complete rewriting systems and homological finiteness properties is given by a result of Anick (1986) which shows that a monoid that admits such a presentation must be of type FPn for all n. For groups, Fn and FPn are equivalent for n = 0,1, while results of Bestvina and Brady (1997) show that FP2 is definitely weaker than F2. For higher n there are no further differences, and in general a group G is of type Fn (for n > 1) if and only if it is finitely presented and of type FPn.
Because of the connection with rewriting systems, the the finiteness property FPn for monoids has received a great deal of attention in the literature. It is sometimes easier to establish the topological finiteness properties Fn for a group than the homological finiteness properties FPn, especially if there is a suitable geometry/topological space on which the group acts in a nice way. Currently no theory of Fn exists for monoids. In this talk I will describe some recent joint work with Benjamin Steinberg (City College of New York) which was motivated by the question of developing a useful notion of Fn for monoids. This led us to develop a theory of monoids acting on CW complexes. I will explain the ideas we have developed and some of their applications.
M-CW complexes and topological finiteness properties of monoids Sponsored by the Meyer Fund
Feb. 14, 2017 2pm (MATH 350)
Lie Theory
Richard Green (CU)
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Generalized nil Temperley--Lieb algebras, which are defined by generators and relations, are certain associative algebras arising from Coxeter systems. I will discuss the extent to which it is possible to classify the finite dimensional representations of such algebras. This will involve a review of some important topics in representation theory, including Morita equivalence and representation type.
Representation theory of nil Temperley--Lieb algebras and related algebras II
Feb. 14, 2017 3pm (Math 220)
Functional Analysis
Alex Kumjian (University of Nevada, Reno)
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Let be an amenable locally compact groupoid, and let be a closed subgroup of a locally compact abelian group . Given a -valued -cocycle on , there is a central extension of by that is trivial iff lifts to a -valued cocycle. We prove that is isomorphic to the induced algebra of the natural action of on . We also consider a simple class of examples arising from Cech -cocycles. This is joint work with Marius Ionescu.
Obstructions to Lifting Cocycles on Groupoids and the Associated -Algebras (Part II)
M-CW complexes and topological finiteness properties of monoids