scholarly journals The Bott–Borel–Weil Theorem for direct limit groups

2001 ◽  
Vol 353 (11) ◽  
pp. 4583-4622 ◽  
Author(s):  
Loki Natarajan ◽  
Enriqueta Rodríguez-Carrington ◽  
Joseph A. Wolf
Keyword(s):  
Direct Limit ◽  
Limit Groups ◽  
Weil Theorem

scholarly journals EQUIVARIANT COMPRESSION OF CERTAIN DIRECT LIMIT GROUPS AND AMALGAMATED FREE PRODUCTS

2016 ◽  
Vol 58 (3) ◽  
pp. 739-752
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Finite Index ◽  
Direct Limit ◽  
Free Products ◽  
Limit Groups ◽  
Amalgamated Free Products

AbstractWe give a means of estimating the equivariant compression of a group G in terms of properties of open subgroups Gi ⊂ G whose direct limit is G. Quantifying a result by Gal, we also study the behaviour of the equivariant compression under amalgamated free products G1∗HG2 where H is of finite index in both G1 and G2.

Differentiable structure for direct limit groups

1991 ◽  
Vol 23 (2) ◽  
pp. 99-109 ◽  
Author(s):  
Loki Natarajan ◽  
Enriqueta Rodr�guez-Carrington ◽  
Joseph A. Wolf
Keyword(s):  
Direct Limit ◽  
Differentiable Structure ◽  
Limit Groups

scholarly journals A Bott–Borel–Weil Theorem for Diagonal Ind-groups

2011 ◽  
Vol 63 (6) ◽  
pp. 1307-1327 ◽  
Author(s):  
Ivan Dimitrov ◽  
Ivan Penkov
Keyword(s):  
Line Bundle ◽  
Weyl Group ◽  
Cohomology Group ◽  
Direct Limit ◽  
Line Bundles ◽  
Precise Description ◽  
Highest Weight Module ◽  
Highest Weight ◽  
Weil Theorem ◽  
Special Case

AbstractA diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusionas a typical special case. If G is a diagonal ind-group and B ⊂ G is a Borel ind-subgroup, we consider the ind-variety G/B and compute the cohomology H𝓁(G/B,𝒪−λ) of any G-equivariant line bundle 𝒪−λ on G/B. It has been known that, for a generic λ, all cohomology groups of 𝒪−λ vanish, and that a non-generic equivariant line bundle 𝒪−λ has at most one nonzero cohomology group. The new result of this paper is a precise description of when Hj (G/B,𝒪−λ) is nonzero and the proof of the fact that, whenever nonzero, Hj (G/B,𝒪−λ) is a G-module dual to a highest weight module. The main difficulty is in defining an appropriate analog WB of the Weyl group, so that the action of WB on weights of G is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to Hj (G/B,𝒪−λ) is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.

scholarly journals Direct limit groups and the Keesling-Mardešić shape fibration

1980 ◽  
Vol 86 (2) ◽  
pp. 471-476 ◽  
Author(s):  
Kenneth Goodearl ◽  
Thomas Rushing
Keyword(s):  
Direct Limit ◽  
Limit Groups

Vanishing of the first-order cohomology on Olshanski spherical pairs

2021 ◽  
pp. 2150030
Author(s):  
Marouane Rabaoui
Keyword(s):  
Direct Limit ◽  
Unitary Representations ◽  
Limit Groups ◽  
First Order ◽  
Cohomology Space ◽  
Invariant Vectors

In this paper, we study the first-order cohomology space of countable direct limit groups related to Olshanski spherical pairs, relatively to unitary representations which do not have almost invariant vectors. In particular, we prove a variant of Delorme’s vanishing result of the first-order cohomology space for spherical representations of Olshanski spherical pairs.

scholarly journals Direct limit groups do not have small subgroups

2007 ◽  
Vol 154 (6) ◽  
pp. 1126-1133 ◽  
Author(s):  
Helge Glöckner
Keyword(s):  
Direct Limit ◽  
Limit Groups

scholarly journals Quantitative residual properties of Γ-limit groups

2014 ◽  
Vol 24 (02) ◽  
pp. 207-231
Author(s):  
Brent B. Solie
Keyword(s):  
Hyperbolic Group ◽  
Finitely Generated ◽  
Limit Group ◽  
Limit Groups ◽  
Residual Properties

Let Γ be a fixed hyperbolic group. The Γ-limit groups of Sela are exactly the finitely generated, fully residually Γ groups. We introduce a new invariant of Γ-limit groups called Γ-discriminating complexity. We further show that the Γ-discriminating complexity of any Γ-limit group is asymptotically dominated by a polynomial.

scholarly journals There are genus one curves of every index over every infinite, finitely generated field

2019 ◽  
Vol 2019 (749) ◽  
pp. 65-86
Author(s):  
Pete L. Clark ◽  
Allan Lacy
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Finitely Generated ◽  
Weil Theorem ◽  
Genus One

Abstract We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E_{/K} which admits infinitely many torsors with index any given n>1 .

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