On Freiman's Theorem in Nilpotent Groups
We generalize a result of Tao which extends Freiman's theorem to the Heisenberg group. We extend it to simply connected nilpotent Lie groups of arbitrary step.
math.CO↗
arXiv subjects
David Fisher
Publications and source records attributed to David Fisher.
Groups acting on manifolds: around the Zimmer program
This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate the current state of the art on as many of these conjectures and questions as possible and $(3)$ to indicate a wide variety of open problems and directions of research.
math.DS↗
Groups not acting on manifolds
In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property.
math.DS↗
Local Rigidity of group actions: Past, Present, Future
This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research. {\em To Anatole Katok on the occasion of his 60th birthday.}
math.DS↗
Strengthening Kazhdan's Property $(T)$ by Bochner Methods
In this paper, we propose a property which is a natural generalization of Kazhdan's property $(T)$ and prove that many, but not all, groups with property $(T)$ also have this property. Let $\G$ be a finitely generated group. One definition of $\G$ having property $(T)$ is that $H^1(\G,π,\fh)=0$ where the coefficient module $\fh$ is a Hilbert space and $π$ is a unitary representation of $\G$ on $\fh$. Here we allow more general coefficients and say that $\G$ has property $F \otimes H$ if $H^1(\G,π_1{\otimes}π_2,F{\otimes}\fh)=0$ if $(F,π_1)$ is any representation with $\dim(F)<\infty$ and $(\fh,π_2)$ is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property $F \otimes H$ if and only if it has property $(T)$. The proof hinges on an extension of a Bochner-type formula due to Matsushima-Murakami and Raghunathan. We give a new and more transparent derivation of this formula as the difference of two classical Weitzenböck formula's for two different structures on the same bundle. Our Bochner-type formula is also used in our work on harmonic maps into continuum products \cite{Fisher-Hitchman2,Fisher-Hitchman1}. Some further applications of property $F \otimes H$ in the context of group actions will be given in \cite{Fisher-Hitchman3}.
math.DG↗
$\Out(F_n)$ and the spectral gap conjecture
For $n>2$, given $ϕ_1,...,ϕ_n$ randomly chosen isometries of $S^2$, it is well-known that the group $\G$ generated by $ϕ_1,...,ϕ_n$ acts ergodically on $S^2$. It is conjectured in \cite{GJS} that for almost every choice of $ϕ_1,...,ϕ_n$ this action is {\em strongly ergodic}. This is equivalent to the spectrum of $ϕ_1+ϕ_1{\inv}+{...}+ϕ_n+ϕ_n^{\inv}$ as an operator on $L^2(S^2)$ having a spectral gap, i.e. all eigenvalues but the largest one being bounded above by some $λ_1<2n$. (The largest eigenvalue $λ_0$, corresponding to constant functions, is $2n$.) In this article we show that if $n>2$, then either the conjecture is true or almost every $n$-tuple fails to have a gap. In fact, the same result is holds for any $n$-tuple $ϕ_1,..., ϕ_n$ in any any compact group $K$ that is an almost direct product of SU(2) factors with $L^2(S^2)$ replaced by $L^2(X)$ where $X$ is any homogeneous $K$ space. A weaker result is proven for $n=2$ and some conditional results for similar actions of $F_n$ on homogeneous spaces for more general compact groups.
math.GR↗
Deformations of group actions
Let $G$ be a noncompact real algebraic group and $\G<G$ a lattice. One purpose of this paper is to show that there is an smooth, volume preserving, mixing action of $G$ or $\G$ on a compact manifold which admits a smooth deformation. We also describe some other, rather special, deformations when $G=SO(1,n)$ and provide a simple proof that any action of a compact Lie group is locally rigid.
math.DS↗
Quasi-isometries and rigidity of solvable groups
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as "coarse differentiation".
math.GR↗
Cocycle superrigidity and harmonic maps with infinite dimensional targets
We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property $(T)$. In particular, our results apply to SP(1,n) and $F_4^{-20}$ and lattices in those groups. The main technical step is to prove a very general result concerning existence of harmonic maps into infinite dimensional spaces, namely a class of simply connected, homogeneous, aspherical Hilbert manifolds. This builds on previous work of Corlette-Zimmer and Korevaar-Schoen. Our result is new because we consider more general targets than previous authors and make no assumption concerning the action. In particular, the target space is not assumed to have non-positive curvature and in important cases has significant positive curvature. We also do not make a "reductivity" assumption concerning absence of fixed points at infinity. The proof of cocycle superrigidity given here, unlike previous ergodic theoretic ones, is effective. The straightening of the cocycle is explicitly a limit of a heat flow. This explicit construction should yield further applications.
math.DG↗
Almost isometric actions, property $(T)$, and local rigidity
Let $Γ$ be a discrete group with property $(T)$ of Kazhdan. We prove that any Riemannian isometric action of $Γ$ on a compact manifold $X$ is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in \cite{FM2}. One definition of property $(T)$ is that a group $Γ$ has property $(T)$ if every isometric $Γ$ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with property $(T)$. Some of these are used in the proofs of our local rigidity theorems.
math.DS↗
Local rigidity of affine actions of higher rank groups and lattices
Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral automorphisms is locally rigid. More generally, given a manifold $M$ on which $Γ$ acts isometrically and a torus $\Ta^n$ on which it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$ is locally rigid. This paper is the culmination of a series of papers and depends heavily on our work in \cite{FM1,FM2}. The reader willing to accept the main results of those papers as "black boxes" should be able to read the present paper without referring to them.
math.DS↗
Continuous quotients for lattice actions on compact manifolds
Let G be a subgroup of finite index in SL(n,Z) for N > 4. Suppose G acts continuously on a manifold M, with fundamental group Z^n, preserving a measure that is positive on open sets. Further assume that the induced G action on H^1(M) is non-trivial. We show there exists a finite index subgroup G' of G and a G' equivariant continuous map from M to the n-torus that induces an isomorphism on fundamental groups. We prove more general results providing continuous quotients in cases where the fundamental group of M surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with G actions to which the theorems apply.
math.DS↗
Nonexistence of invariant rigid structures and invariant almost rigid structures
We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spaces. The actions on homogeneous spaces all preserve affine connections, whereas the action along the exceptional divisor preserves a projective structure. The fact that these structures cannot in some way be "glued together" to give a rigid structure on the entire space is not obvious. We also define the notion of an almost rigid structure. The paradigmatic example of a rigid structure is a global framing and the paradigmatic example of an almost rigid structure is a framing that is degenerate along some exceptional divisor. We show that the actions discussed above do possess an invariant almost rigid structure. Gromov has shown that a manifold with rigid geometric structure invariant under a topologically transitive group action is homogeneous on an open dense set. How generally this open dense set can be taken to be the entire manifold is an important question with many dynamical applications. Our results indicate one way in which the geometric structure cannot degenerate off the open dense set.
math.DS↗
Local Rigidity for Cocycles
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a continuous perturbation of a constant cocycle is actually continuously conjugate back to the original constant cocycle modulo a cocycle that is continuous and "small". We give some applications to perturbations of standard actions of higher rank semisimple Lie groups and their lattices. Some of the results proven here are used in our proof of local rigidity for affine and quasi-affine actions of these groups. We also improve and extend the statements and proofs of Zimmer's cocycle superrigidity.
math.DS↗
When is a group action determined by its orbit structure?
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit equivalence is equivariant. Even in the presence of larger fixed sets, for actions preserving rigid geometric structures our results force sufficiently smooth orbit equivalences to be equivariant. For instance, if a countable group $Γ$ acts on ${\mathbb T}^n$ and the action is $C^1$ orbit equivalent to the standard action of $SL_n({\mathbb Z})$ on ${\mathbb T}^n$, then $Γ$ is isomorphic to $SL_n({\mathbb Z})$ and the actions are isomorphic. (The same result holds if we replace $SL_n(\mathbb Z)$ by a finite index subgroup.) We also show that preserving a geometric structure is an invariant of smooth orbit equivalence and give an application of our ideas to the theory of hyperbolic groups. In the course of proving our theorems, we generalize a theorem of Sierpinski which says that a connected Hausdorff compact topological space is not the disjoint union of countably many closed sets. We prove a stronger statement that allows "small" intersections provided the space is locally connected. This implies that for any continuous action of a countable group $\G$ on a connected, locally connected, locally compact, Hausdorff topological space, where the fixed set of every element is "small", every orbit equivalence is equivariant.
math.DS↗
The Color-Magnitude Relation in CL 1358+62 at z=0.33: Evidence for Significant Evolution in the S0 Population
We use a large mosaic of HST WFPC2 images to measure the colors and morphologies of 194 spectroscopically confirmed members of the rich galaxy cluster CL1358+62 at z=0.33. We study the color-magnitude (CM) relation as a function of radius in the cluster. The intrinsic scatter in the restframe B-V CM relation of the elliptical galaxies is very small: ~0.022 magnitudes. The CM relation of the ellipticals does not depend significantly on the distance from the cluster center. In contrast, the CM relation for the S0 galaxies does depend on radius: the S0s in the core follow a CM relation similar to the ellipticals, but at large radii (R>0.7Mpc) the S0s are systematically bluer and the scatter in the CM relation approximately doubles to ~0.043 magnitudes. The blueing of the S0s is significant at the 95% confidence level. These results imply that the S0 galaxies in the outer parts of the cluster have formed stars more recently than the S0s in the inner parts. A likely explanation is that clusters at z=0.33 continue to accrete galaxies and groups from the field and that infall extinguishes star formation. The apparent homogeneity of the elliptical galaxy population implies that star formation in recently accreted ellipticals was terminated well before accretion occurred. We have constructed models to explore the constraints that these observations place on the star formation history of cluster galaxies. We conclude that the population of S0s in clusters is likely to evolve as star forming galaxies are converted into passively evolving galaxies. Assuming a constant accretion rate after z=0.33, we estimate ~15% of the present day early-type galaxy population in rich clusters was accreted between z=0.33 and z=0. The ellipticals (and the brightest S0s) are probably a more stable population, at least since z=0.6.
astro-ph↗
A Search for Counter-Rotating Stars in S0 Galaxies
We have obtained high signal-to-noise spectra along the major axes of 28 S0 galaxies in order to search for the presence of disk stars on retrograde orbits. Full line-of-sight velocity distributions were extracted from the data, and the velocity distributions were modelled as arising from the superposition of populations of stars on prograde and retrograde orbits. We find no new cases in which a significant fraction of disk stars lie on retrograde orbits; an identical analysis of NGC~4550 does reveal the previously-known counter-rotating stellar disk in this system. Upper limits determined for each object indicate that no more than about 5% of the observed disk star light could arise from counter-rotating stellar components. These results suggest that previously-discovered disk galaxies with counter-rotating stars are exceptional and that (at 95% confidence) at most 10% of S0 galaxies contain significant counter-rotating populations. The most likely value for the fraction of such S0 galaxies lies closer to 1%. This result contrasts with the prevalence of counter-rotating gas in these systems; combining our new observations with existing data, we find that 24+/-8% (1-sigma error) of the gas disks in S0 galaxies counter-rotate relative to their stellar components.
astro-ph↗
CRYSTAL: Inducing a Conceptual Dictionary
One of the central knowledge sources of an information extraction system is a dictionary of linguistic patterns that can be used to identify the conceptual content of a text. This paper describes CRYSTAL, a system which automatically induces a dictionary of "concept-node definitions" sufficient to identify relevant information from a training corpus. Each of these concept-node definitions is generalized as far as possible without producing errors, so that a minimum number of dictionary entries cover the positive training instances. Because it tests the accuracy of each proposed definition, CRYSTAL can often surpass human intuitions in creating reliable extraction rules.
cmp-lg↗