The action of nilpotent groups on infinite graphs

First we prove a result about the action of nilpotent groups on the set of ends of locally finite graphs. This theorem has immediate consequences for the structure of graphs which allow a transitive action of those groups. Further we investigate the cycle-structure of automorphisms of a transitive nilpotent group and the existence of abelian groups acting on sets of imprimitivity of graphs whose automorphism groups have transitive nilpotent subgroups.     

The action of Young subgroups on the partition complex

Publications mathématiques de l'IHÉS - We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_{n}|$ , which is the $\Sigma_{n}$...     

The action of

We prove that the orbit equivalence relation of the canonical action of on the subsets of is a universal countable Borel equivalence relation.     

The analysis of popular collective action

Most models of popular collective action treat it as the behavior of single social units having unitary dispositions that explain the behavior. Since in fact it consists largely of strategic interaction among several parties, activates and builds on existing social networks, and follows a dynamic that no single-actor model can represent, available models are inadequate. Through sustained discussion of the rural conflicts of 1830 in England, this paper illustrates the critique, sketches an alternative approach centering on the analysis of repertoires of contention, and describes techniques for standardizing narratives of contention that are compatible with the alternative approach.     

The fixed points of coprime action

Let P be an odd p\pi -group that acts as a group of automorphisms on the soluble p￠\pi '-group G. We obtain generators for the fixed points of P on [G, P].     

The action of a few permutations on r-tuples is quickly transitive

We prove that for every r and d≥2 there is a C such that for most choices of d permutations π₁, π₂,…,π_d of S_n, the following holds: for any two r-tuples of distinct elements in {1,…,n}, there is a product of less than C log n of the π_is which map the first r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to a general context of which random Cayley graph quotients of S_n are good expanders. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 335–350, 1998     

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.     

The isotropy lattice of a lifted action

We obtain a characterization of the isotropy lattice for the lifted action of a Lie group G on TM and $T^{1}M$ based only on the knowledge of G and its action on M. Some applications to symplectic geometry are also shown. To cite this article: M. Rodríguez-Olmos, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The number of exceptional orbits of a pseudofree circle action on S 5

Let M be a closed Riemannian manifold of dimension 5 which admits a Riemannian metric of nonnegative sectional curvature. The aim of this short paper is to show that under certain lower bound of the orders of isotropy subgroups, every pseudofree and isometric S ¹-action on M cannot have more than five exceptional circle orbits. As a consequence, we conclude that a pseudofree and isometric S ¹-action on a 5-sphere S ⁵ with a Riemannian metric of nonnegative sectional curvature cannot have more than five exceptional circle orbits. This gives a result related to the Montgomery–Yang problem. In addition, we also give some further related result about nonnegatively curved manifolds of dimension 5 with an isometric but not necessarily pseudofree circle action.     

The symplectic normal space of a cotangent-lifted action

For the cotangent bundle T^∗Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.     

The unregularized gradient flow of the symplectic action

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.