Our aim is to explore and understand the rigidity and exibility of these actions. The papers in geometry, rigidity, and group actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In other words, up to finite index, all geometric actions on s1 come from surface groups. The new ingredient is a dynamical cocycle super rigidity theorem. The author is grateful to the organizers of the rims conferences where these results were. We prove a rigidity theorem for the coarse geometry of such warped cones. Neumann rigidity of cusps in deformations of hyperbolic 3orbifolds. Global rigidity of solvable group actions on s1 arxiv. Steklova, fjournal trudy matematicheskogo instituta imeni v. Integrable systems and group actions 3 the symplectic geometry and topology of these singularities 21, 20, 47, 6, 79, and more recently the work in connection to semitoric actions and singularities 66, 67.
A survey of measured group theory in geometry, rigidity, and group actions 296 374. Spitler on the profinite rigidity of triangle groups. Quiver representations, group characters, and prime graphs of finite groups iiyori, nobuo and sawabe, masato, tokyo journal of mathematics, 2019. In discrete groups and geometry, proceedings in honour of the retirement of a. Then each group quasiisometricto mustbevirtuallyisomorphicto. Our primary focus is on actions of surface groups, with the aim of introducing the reader to recent developments and new tools to study groups acting by. Groups acting on the circle rigidity, exibility, and moduli spaces of actions kathryn mann uc berkeley msri nitely generated group. July 19, 2000 1 introduction let m be a smooth manifold, g a geometric structure on m, and ga lie group that acts on m so as to leave g invariant. Proceedings of the georgia international topology conference 18. Group actions math 415b515b the notion of a group acting on a set is one which links abstract algebra to nearly every branch of mathematics.
The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. In these actions, elliptic and hyperbolic dynamics coexist. Rigidity and flexibility of group actions on the circle uc berkeley math. He is the author of problems on mapping class groups and related topics and coauthor of. This led naturally to a rigidity result for quasiconvex geometric actions on cat. Peter may 1967, 1993 fields and rings, second edition, by irving. The course introduced a number of classical tools in smooth ergodic theory particularly lyapunov exponents and metric entropy as tools to study rigidity properties of group actions on manifolds. We rst show, using the kam method, that any small and su ciently smooth pertur. The geometry of synchronization problems and learning group actions article pdf available in discrete and computational geometry october 2016 with 119 reads how we measure reads.
We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. The new ingredient is a dynamical cocycle superrigidity theorem. Ergodic theory, group representations, and rigidity. The notion of quasiisometry is basic in geometric group theory.
In this article we prove global rigidity results for hyperbolic actions of higherrank lattices. In the same chapter we use similar zooming arguments to prove the special case of mostow rigidity theorem. An ergodic theorem for the quasiregular representation of the free group boyer, adrien and lobos, antoine pinochet, bulletin of the belgian mathematical society simon stevin, 2017. Rigidity of some abelianbycyclic solvable group actions on tn amie wilkinson, jinxin xue abstract. For this reason we will study them for a bit while taking a break from ring theory. Rigidity of pseudofree group actions on contractible. Rigidity and exibility of group actions on the circle kathryn mann abstract we survey rigidity results for groups acting on the circle in various settings, from local to global and c0 to smooth. Combinatorial use edit in combinatorics, the term rigid is also used to define the notion of a rigid surjection, which is a surjection f. Past, present, future david fisher to anatole katok on the occasion of his 60th birthday. The author is grateful to the organizers of the rims conferences where these results. Group actions appear in geometry, linear algebra, and di erential equations, to name a few. Geometry, rigidity, and group actions, farb, fisher. Rigidity problems for integrable systems in these manifolds will be explored from this perspective. Systems 16, 1996 and recent the work of the main author with d.
Global smooth and topological rigidity of hyperbolic lattice actions aaron brown, federico rodriguez hertz, and zhiren wang abstract. Geometry, rigidity, and group actions edited by benson farb and david fisher the university of chicago press chicago and london benson farb is professor of mathematics at the university of chicago. Rigidity for quasimobius actions on fractal metric spaces kinneberg, kyle, journal of differential geometry, 2015. Global smooth and topological rigidity of hyperbolic. The subject of these lectures will be the general interplay between the dynamics of the gaction. One of the earliest and most in uential results in the area in fact a precursor to the eld of geometric group theory is mostows. There is a particularly detailed discussion of recent results, including outlines of some proofs. In this paper, we study a natural class of groups that act as a ne trans. Entropy, lyapunov exponents, and rigidity of group actions. Moment maps, cobordisms, and hamiltonian group actions. Rigidity, dynamics, and group actions david fisher lehman college cuny, elon lindenstrauss princeton university, dave witte morris university of lethbridge, ralf spatzier university of michigan. Title differential rigidity of anosov actions of higher rank abelian groups and algebraic lattice actions, journal tr. Workshop on \rigidity, group actions, and dynamics birs, ban, canada, coorganized with elon lindenstrauss, dave witte morris and ralf spatzier, july 914, 2005.
Geometry, rigidity, and group actions pdf free download. Groups acting on the circle rigidity, exibility, and moduli spaces of actions. Some of this material is covered in chapter 7 of gallians book, but we will take a slightly more general approach. Rigidity and exibility of group actions on the circle. Examples of amalgamated free products and coupling rigidity volume 33 issue 2 yoshikata kida. This survey aims to cover the motivation for and history of the study of local rigidity of group actions. Groups of uniform homeomorphisms of covering spaces yagasaki, tatsuhiko, journal of the mathematical society of japan, 2014. Rigid geometric structures and actions of semisimple lie. Library of congress cataloginginpublication data guillemin, v. The notion of integrability and its connection to group actions can be naturally studied in the contact context too. The study of local rigidity of lattices in semisimple lie groups is probably the beginning of the general study of rigidity in geometry and dynamics, a subject. Orbitequivalence rigidity for higherrank lattice subgroups 68 10d. Groups, geometry, and rigidity kathryn mann notes from a minicourse given at mit, march 2017 abstract this minicourse is an introduction to some central themes in geometric group theory and their modern o shoots.
May other chicago lectures in mathematics titles available from the university of chicago press simplical objects in algebraic topology, by j. Hyperbolic dynamics, invariant geometric structures and. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. The question of quasiisometric rigidity of a group. Rather, we focus on two rigidity results in higherrank dynamics. We do not present comprehensive treatment of group actions or general rigidity programs. Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces. This includes banach space property t, the xed point property for. The discrete, cocompact subgroups of so2 are exactly the finite. Long embedding closed hyperbolic 3manifolds in small volume hyperbolic 4manifolds. A characterization of fuchsian actions by topological rigidity. Handbook of group actions international press of boston.
This is also closely related to work of katokspatzier discussed below in 2. A survey of measured group theory in geometry, rigidity, and group actions 296 374, the university of chicago press, chicago and london, 2011. Msc 2010 20f65 20f67 20f69 20f05 20f10 20e08 20e05 20e06 geometric group theory group actions and geometry quasiisometry of groups cayley graphs of groups rigidity in group theory curvature and fundamental groups hyperbolic groups negatively curved groups amenable groups growth of groups gromov boundary. Pdf the geometry of synchronization problems and learning. Hyperbolic dynamics, invariant geometric structures and rigidity of abelian group actions march 16, 2004 these notes written by v. Global smooth and topological rigidity of hyperbolic lattice. The main purpose of this paper is to present in a uni ed approach di erent results concerning group actions and integrable systems in symplectic, poisson and contact manifolds. We will present a proof of this theorem in chapter 22. Examples of amalgamated free products and coupling rigidity. Group actions, geometry, and rigidity kathryn mann, brown university page 1253 numerical algebraic geometry and optimization jonathan d. To appear in proceedings of the georgia international topology conference 18. Salgueiro some remarks on group actions on hyperbolic 3manifolds.
Pdf rigidity theorems for higher rank lattice actions. The book can be used by researchers and graduate students working in symplectic geometry and its applications. In this paper, we study the rigidity properties of a class of a ne solvable group actions on tn. The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. Rigidity of group actions on homogeneous spaces, iii. We focus on two perspectives that have particularly strong. The asymptotic geometry of negatively curved spaces. Geometry, rigidity, and group actions chicago lectures in. Katok at the pennsylvania state university in the spring of 1994. Geometry, rigidity, and group actions cern document server.
July 9 july 14, 2005 rigidity theory has its roots in classical theorems of selberg, weil, mostow, margulis and furstenberg. We begin by motivating the study of spaces of group actions, as well as our focus on surface groups. Hauenstein, university of notre dame page 1251 sampler. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semisimple lie groups. This is a survey of recent developments at the interface between quasiconformal analysis and the asymptotic geometry of gromov hyperbolic groups. Surface group representations with maximal toledo invariant pdf, 1. Namely, if a group has no abelian factors, then two such warped cones are quasiisometric if and only if the actions are nite covers of. Geometry, rigidity, and group actions chicago lectures in mathematics series editors. Approximate equivalence of group actions ergodic theory and.
An extension criterion for lattice actions on the circle pdf, 222 kb geometry, rigidity, and group actions, 331, chicago lectures in math. In particular, suppose x is a locally finite cat0 polyhedral complex and. Geometry, rigidity, and group actions will appeal to a wide range of mathematicians. Rigidity of group actions on homogeneous spaces, iii bader, uri, furman, alex, gorodnik, alex, and weiss, barak, duke mathematical journal, 2015. The book geometry, rigidity, and group actions, edited by benson farb and david fisher is published by university of chicago press. For example, local rigidity of the higher rank abelian actions led to the proof of local rigidity of projective actions of higher rank cocompact lattices. On the fundamental group of ii 1 factors and equivalence relations arising from group actions. The papers in geometry, rigidity, and group actions explore the role of group actions and rigidity in several areas. Suppose is a lattice in semisimple lie group, all of whose factors have rank 2 or higher. Local rigidity of affine actions of higher rank groups and. Tatsuki seto nagoya toeplitz operators and the roehigson type index theorem. Rigidity results in ktheory show isomorphisms between various algebraic ktheory groups. Lectures on geometric group theory uc davis mathematics. Approximate equivalence of group actions ergodic theory.
More recent developments concerning global rigidity of group actions have introduced a plethora of new. Rigidity of abc solvable group actions 3 all such groups are of the form b, where b b ij is an integer valued, d d matrix with detb 6 0, and 1. Apr 30, 2011 the papers in geometry, rigidity, and group actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. Integrable systems and group actions eva miranda abstract. If a group acts by isometries on x with a bounded orbit, then has a fixed point in x. The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic.
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