Stabilizers of actions of lattices in products of groups

2016 ◽  
Vol 37 (4) ◽  
pp. 1133-1186 ◽  
Author(s):  
DARREN CREUTZ
Keyword(s):  
Simple Groups ◽  
Ergodic Decomposition ◽  
Semisimple Groups ◽  
Products Of Groups ◽  
Algebraic Properties ◽  
Simple Factor ◽  
Simple Subgroup ◽  
Contractive Maps ◽  
Higher Rank ◽  
Invariant Random Subgroups

We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups. Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having property $(T)$, which is ergodic for each simple subgroup, is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe–Moore property, rather than relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.

Stabilizers for Ergodic Actions of Higher Rank Semisimple Groups

1994 ◽  
Vol 139 (3) ◽  
pp. 723 ◽  
Author(s):  
Garrett Stuck ◽  
Robert J. Zimmer
Keyword(s):  
Semisimple Groups ◽  
Higher Rank

Finitely generated profinitely dense free groups in higher rank semi-simple groups

2000 ◽  
Vol 5 (1) ◽  
pp. 93-100 ◽  
Author(s):  
G. A. Soifer ◽  
T. N. Venkataramana
Keyword(s):  
Free Groups ◽  
Simple Groups ◽  
Finitely Generated ◽  
Higher Rank

scholarly journals On the Growth of L2-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One

2018 ◽  
Vol 2020 (9) ◽  
pp. 2588-2625
Author(s):  
Miklos Abert ◽  
Nicolas Bergeron ◽  
Ian Biringer ◽  
Tsachik Gelander ◽  
Nikolay Nikolov ◽  
...  
Keyword(s):  
Asymptotic Behavior ◽  
Lie Groups ◽  
Symmetric Spaces ◽  
Betti Numbers ◽  
Real Rank ◽  
Spectral Invariants ◽  
Locally Symmetric Spaces ◽  
Rank One ◽  
Higher Rank ◽  
Invariant Random Subgroups

Abstract In the 1st paper of this series we studied the asymptotic behavior of Betti numbers, twisted torsion, and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo–Stuck–Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statements about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n, 1), n ≥ 2. We give dimension-specific constructions when n = 2, 3, and also describe a general gluing construction that works for every n. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro’s construction of non-arithmetic lattices in SO(n, 1).

Groups with many characteristically simple subgroups

1979 ◽  
Vol 86 (2) ◽  
pp. 193-198 ◽  
Author(s):  
J. S. Wilson
Keyword(s):  
The Other ◽  
Simple Groups ◽  
Countable Subgroup ◽  
Simple Subgroup ◽  
Similar Assertion ◽  
Elementary Result

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

On the Simple Group of J. Tits

1973 ◽  
Vol 16 (1) ◽  
pp. 87-92
Author(s):  
David Parrott
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Simple Subgroup ◽  
Index 2

In the series of simple groups 2F4(q),q =2 2m+1, discovered by Ree, Tits [4] showed that the group 2F4(2) was not simple but contained a simple subgroup of index 2. In this note we extend the characterization of obtained by the author in [3].

scholarly journals A formal analysis of some factor- and precedent-based accounts of precedential constraint

2021 ◽  
Author(s):  
Henry Prakken
Keyword(s):  
Formal Analysis ◽  
Simple Factor ◽  
The Difference

AbstractIn this paper several recent factor- and dimension-based models of precedential constraint are formally investigated and an alternative dimension-based model is proposed. Simple factor- and dimension-based syntactic criteria are identified for checking whether a decision in a new case is forced, in terms of the relevant differences between a precedent and a new case, and the difference between absence of factors and negated factors in factor-based models is investigated. Then Horty’s and Rigoni’s recent dimension-based models of precedential constraint are critically examined. An alternative to their reason models is proposed which is less expressive but arguably easier to apply in practice.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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