Dipendra Prasad
These are the notes for some lectures given by this author at Harish-Chandra Research Institute, Allahabad in March 2014 for a workshop on Schur multipliers. The lectures aimed at giving an overview of the subject with emphasis on groups of Lie type over finite, real and $p$-adic fields.
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The purpose of this note is to give a number of open problems on matching theory and their relation to the well-known results in this area. We also give a linear analogue of the acyclic matchings.
P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$.
We consider central extensions $Z\hookrightarrow E\twoheadrightarrow G$ in the category of linear differential algebraic groups. We show that if $G$ is simple non-commutative and $Z$ is unipotent with the differential type smaller than that of $G$, then such an extension splits. We also give a construction of central extensions illustrating that the condition on differential types is important for splitting. Our results imply that non-commutative almost simple linear differential algebraic groups, introduced by Cassidy and Singer, are simple.
On the isomorphism problem for central extensions II
本文研究中心扩张的同构问题，在商群为单群或纯非阿贝尔群等新情形下，给出了两个中心扩张同构的充分必要条件，并刻画了保持商群不变及核群与商群同构时的同构问题。
We construct a central Lie group extension for the Lie group of compactly supported sections of a Lie group bundle over a sigma-compact base manifold. This generalises a result of the paper "Central extensions of groups of sections" by Neeb and Wockel, where the base manifold is assumed to be compact. In the second part of the paper, we show that this extension is universal and obtain a generalisation of a corresponding result in the paper "Universal central extensions of gauge algebras and groups" by Janssens and Wockel, where again (in the case of Lie group extensions) the base manifold is a…
We find a necessary condition for the embedding of a central extension of a group $G$ with elementary abelian kernel into the wreath product that corresponds to a permutation action of $G$. The proof uses purely group-theoretic methods.
The Equation Problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions of hyperbolic groups is solvable.
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integra…
We find a criterion for the embedding of a nonsplit central extension of $PSL_n(q)$ with kernel of prime order into the permutation wreath product that corresponds to the action on the projective space.
It is shown that FC-central extensions retain sub-exponential volume growth. A large collection of FC-central extensions of the first Grigorchuk group is provided by the constructions in the works of Erschler and Kassabov-Pak. We show that in these examples subgroup separability is preserved. We introduce two new collections of extensions of the Grigorchuk group. One collection gives first examples of intermediate growth groups with centers isomorphic to $\mathbb{Z}^{\infty}$; and the other provides groups with prescribed oscillating intermediate growth functions.
We show that every nilpotent group of class at most two may be embedded in a central extension of abelian groups with bilinear cocycle. The embedding is shown to depend only on the base group. Some refinements are obtained by considering the cohomological situation explicitly.
Mickelsson defined a group 2-cocycle on the group of smooth maps from the closed unit 2-disk to a Lie group G and constructed a smooth central extension on a loop group of G, called the affine Kac-Moody central extension. We reformulate this central extension theory in the terms of the seven-term exact sequence for cohomology groups for discrete groups.
The construction of families of Sato Grassmannians, their determinant line bundles and the extensions induced by them are given. The base scheme is an arbitrary scheme.
We define measures on central extension of current groups in any dimension by using infinite dimensional Brownian motion.
We use topological methods to compute the mod p cohomology of certain p-groups. More precisely we look at central Frattini extensions of elementary abelian by elementary abelian groups such that their defining k-invariants span the entire image of the Bockstein. We show that if p is sufficiently large, then the mod p cohomology of the extension can be explicitly computed as an algebra.
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