Some new rigidity results for stable orbit equivalence

1995 ◽  
Vol 15 (2) ◽  
pp. 209-219 ◽  
Author(s):  
Scot Adams
Keyword(s):  
Hyperbolic Group ◽  
Finite Measure ◽  
Universal Cover ◽  
The Other ◽  
Stable Orbit ◽  
Rigidity Result ◽  
Rigidity Results ◽  
Word Hyperbolic Group ◽  
Orbit Equivalent ◽  
Rank One

AbstractBroadly speaking, we prove that an action of a group with very little commutativity cannot be stably orbit equivalent to an action of a group with enough commutativity, assuming both actions are free and finite measure preserving. For example, one group may be SL2(ℝ) and the other a group with infinite discrete center (e.g., the universal cover of SL2(ℝ)); I believe this is the first rigidity result of this type for a pair of simpleLie groups both of split rank one. Another example: one group may be any nonelementary word hyperbolic group, the other any group with infinite discrete center.

scholarly journals Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

2018 ◽  
Vol 40 (1) ◽  
pp. 117-141 ◽  
Author(s):  
DANIJELA DAMJANOVIĆ ◽  
DISHENG XU
Keyword(s):  
Fixed Point ◽  
Universal Cover ◽  
Diffeomorphism Group ◽  
Finite Cover ◽  
Diffeomorphism Groups ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Rigidity Results ◽  
Anosov Actions ◽  
Higher Rank

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Moon Lake’s Orphans and “The Other Way to Live”

2019 ◽  
pp. 79-102
Author(s):  
Jean C. Griffith
Keyword(s):  
Twentieth Century ◽  
Child Welfare ◽  
Welfare Policy ◽  
The Other ◽  
American Democracy ◽  
Child Welfare Policy ◽  
Early Twentieth Century ◽  
Rank One ◽  
Poor Children ◽  
Moon Lake

This essay examines the roles the character Easter in “Moon Lake” plays in the context of early-twentieth-century debates about the roots of poverty and society’s level of responsibility to poor children. By placing the focus of the story not on Easter but on the genteel Morgana girls’ shifting attitudes about her, Welty illustrates the ways child welfare policy was shaped by conflicting attitudes, whereby sympathy for innocent children coexisted with scorn for their parents. Assuming that Easter lives outside the boundaries that mark their own places in Morgana’s gendered, class-bound, and racially-segregated society, Jinny Love Stark and Nina Carmichael imagine the “orphan” to embody a womanhood untethered by race or rank, one, perhaps, more representative of American democracy. Ultimately, though, the girls come to see that Easter’s status as an orphan makes her more marked by and vulnerable to the violence and oppression that shape the South’s racial patriarchy.

scholarly journals Cocycle rigidity of partially hyperbolic abelian actions with almost rank-one factors

2017 ◽  
Vol 39 (7) ◽  
pp. 2006-2016
Author(s):  
KURT VINHAGE
Keyword(s):  
Recent Progress ◽  
Universal Cover ◽  
Cycle Structure ◽  
Rank One ◽  
Partially Hyperbolic ◽  
Cocycle Rigidity

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank-one factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.

scholarly journals Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

2015 ◽  
Vol 07 (03) ◽  
pp. 407-451 ◽  
Author(s):  
Urs Frauenfelder ◽  
Clémence Labrousse ◽  
Felix Schlenk
Keyword(s):  
Lower Bound ◽  
Polynomial Growth ◽  
Polynomial Complexity ◽  
Cohomology Ring ◽  
Volume Growth ◽  
Universal Cover ◽  
Geodesic Flows ◽  
Dynamical Complexity ◽  
Reeb Flows ◽  
Rank One

We give a uniform lower bound for the polynomial complexity of Reeb flows on the spherization (S*M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our lower bound is in terms of the polynomial growth of the homology of the based loop space of M. As an application, we extend the Bott–Samelson theorem from geodesic flows to Reeb flows: If (S*M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fiber [Formula: see text], then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M agrees with that of a compact rank one symmetric space.

The Savoyard Cousins: A Comparison of the Careers and Relative Success of the Grandson (Grandison) and Champvent (Chavent) Families in England

2006 ◽  
Vol 86 ◽  
pp. 148-178
Author(s):  
Michael Ray
Keyword(s):  
Thirteenth Century ◽  
The Other ◽  
Royal Patronage ◽  
Relative Success ◽  
Rank One ◽  
Family Based

In the mid-thirteenth century members of two branches of a family based in Savoy came to England and, through royal service, they reached baronial rank. One family, the Grandsons, thoroughly embedded itself in England and its members are recalled even today while the other, the Champvents, lapsed into obscurity, the name disappearing from the records after 1410. To discover why, this article looks at the significance of royal service to the families, the amount of royal patronage they received, their marriage strategies, how they related to the localities into which they were implanted, the extent to which religious loyalties and family piety illustrated their attitudes and whether they cut their ties with their former home lands.

scholarly journals RANK ONE CONNECTIONS ON ABELIAN VARIETIES, II

2012 ◽  
Vol 23 (12) ◽  
pp. 1250125
Author(s):  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE ◽  
A. K. RAINA
Keyword(s):  
Line Bundle ◽  
Exact Sequence ◽  
Abelian Varieties ◽  
Explicit Construction ◽  
The Other ◽  
Holomorphic Line Bundle ◽  
Canonical Isomorphism ◽  
Complex Torus ◽  
Rank One

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle [Formula: see text], where α is the addition map on A × A, and p1 is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

scholarly journals Symbolic coding for the geodesic flow associated to a word hyperbolic group

2002 ◽  
Vol 109 (4) ◽  
pp. 465-492 ◽  
Author(s):  
Michel Coornaert ◽  
Athanase Papadopoulos
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Group ◽  
Word Hyperbolic Group ◽  
Symbolic Coding

scholarly journals Osserman pseudo-Riemannian manifolds of signature (2,2)

2001 ◽  
Vol 71 (3) ◽  
pp. 367-396 ◽  
Author(s):  
Novica Blažić ◽  
Neda Bokan ◽  
Zoran Rakić
Keyword(s):  
Riemannian Manifold ◽  
Characteristic Polynomial ◽  
Riemannian Manifolds ◽  
Jacobi Operator ◽  
Tangent Vector ◽  
Jordan Form ◽  
The Other ◽  
Jacobi Operators ◽  
Osserman Manifold ◽  
Rank One

AbstractA pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

ENDS OF CUSP-UNIFORM GROUPS OF LOCALLY CONNECTED CONTINUA — I

2005 ◽  
Vol 15 (04) ◽  
pp. 765-798 ◽  
Author(s):  
DAN P. GURALNIK
Keyword(s):  
Hyperbolic Group ◽  
Infinite Group ◽  
Graph Of Groups ◽  
Locally Finite ◽  
Cut Point ◽  
Word Hyperbolic Group ◽  
Complete Knowledge ◽  
Simplicial Tree ◽  
Locally Connected Continuum ◽  
Uniform Group

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

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