Free, Isometric Circle Actions on Compact Symmetric Spaces

Let S=G/K be a strongly irreducible, simply connected, compact symmetric space and let be its group of isometries. We classify the symmetric spaces among these that admit free, isometric circle actions. The existence of such actions is important in constructing examples of manifolds with positive sectional curvature.     

Intersection Form, Laminations and Currents on Free Groups

Let F be a free group of rank N ≥ 2, let μ be a geodesic current on F and let T be an ${\mathbb{R}}Let F be a free group of rank N ≥ 2, let μ be a geodesic current on F and let T be an \mathbbR{\mathbb{R}}-tree with a very small isometric action of F. We prove that the geometric intersection number áT,m?{{\langle T,\mu \rangle}} is equal to zero if and only if the support of μ is contained in the dual algebraic lamination L ²(T) of T. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. We use the main result to obtain “unique ergodicity” type properties for the attracting and repelling fixed points of atoroidal iwip elements of Out(F) when acting both on the compactified outer Space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two “sufficiently transverse” very small F-trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in F and prove that filling elements are “nearly generic” in F. We also apply our results to the notion of bounded translation equivalence in free groups.     

Free Group Actions on Spaces Homotopy Equivalent to a Sphere

For infinite-dimensional homotopy space forms X/G with G nontrivial finite cyclic groups, we study the homotopy type of X/G. We show that the Euler class of X/G is not zero if and only if X/G is finitely dominated.     

Warsaw圈上映射的等度连续性

设W是Warsaw圈：f:W→W是连续映射，本证明f是等度连续映射的充分必要条件是下列两个条件之一成立：（1）F（f)是一个单点集并且F(f^2)=∩^∞n=1f^n(W);(2)F(f)=∩^∞n=1f^n(W)。     

Circle Actions and Morse Theory on Quaternion-Kahler Manifolds

In this paper we study quaternion-Kähler manifolds endowedwith an isometric S¹-action. We consider the corresponding momentmap µ and prove that the only compact quaternion-Kählermanifold with positive scalar curvature which admits an isometriccircle action free on µ^–1(0) is the quaternionicprojective space HPⁿ.     

Combinatorial Harmonic Maps and Discrete-group Actions on Hadamard Spaces

In this paper we use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard spaces, we formulate criterions for the action to have a global fixed point.Dedicated to Professor Takushiro Ochiai on his 60th birthday.     

Actions of Semisimple Lie Groups on Circle Bundles

Suppose G is a connected, simple, real Lie group with -rank(G) 2, M is an ergodic G-space with invariant probability measure , and : G × M Homeo( ) is a Borel cocycle. We use an argument of É. Ghys to show that there is a G-invariant probability measure on the skew product M × , such that the projection of to M is . Furthermore, if (G × M) Diff¹( ), then can be taken to be equivalent to × , where is Lebesgue measure on ; therefore, is cohomologous to a cocycle with values in the isometry group of .     

华沙圈上连续映射的distality与等度连续性

设W为华沙圈,f:W→W为连续映射.本文得到了f为distal的一个刻画并且讨论了f的distality与等度连续性的关系.证明了:(i)f是distal的当且仅当f为恒等映射.(ii)如果f为满射,则f是distal的当且仅当f为等度连续的.     

Bordism-Finiteness and Semi-simple Group Actions

We give bordism-finiteness results for smooth S ³-manifolds. Consider the class of oriented manifolds which admit an S ¹-action with isolated fixed points such that the action extends to an S ³-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontryagin class p ₁, for example p ₁ = 0, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with S ³-action as above.     

Hyperkahler Structures and Group Actions

A 4n-dimensional Riemannian manifold (M, g) is hyperkählerif it possesses three anti-commuting complex structures I, J,K such that the metric g is Kähler with respect to eachof them. The reduced holonomy group of such a manifold is necessarilya subgroup of Sp(n) so the Ricci tensor of g vanishes and (M,g) can be regarded as a positive definite solution to Einstein'sequations in vacuum.     

Group Actions on the Cubic Tree

It is known that every group which acts transitively on the ordered edges of the cubic tree ₃, with finite vertex stabilizer, is isomorphic to one of seven finitely presented subgroups of the full automorphism group of ₃–one of which is the modular group. In this paper a complete answer is given for the question (raised by Djokovi and Miller) as to whether two such subgroups which intersect in the modular group generate their free product with the modular group amalgamated.     

Finite Group Actions on Spin Bundles

Let L be a complex line bundle over a closed, oriented, smooth 4-manifold X with c₁(L) w₂(TX) mod 2. Let a finite group G act on X as orientation preserving isometries and on L such that the projection L X is a G-map. We investigate the action of G on the Seiberg-Witten equations, and when G = Z₂ we study the G-invariant Seiberg-Witten invariants on X and the Seiberg-Witten invariants on its quotient setting.     

Cyclic Group Actions on 4-Manifolds

Let X be a closed, oriented Riemannian 4-manifold. Suppose that a cyclic group Z( _p (p is prime) acts on X by an orientation preserving isometry with an embedded Riemann surface as fixed point set. We study the representation of Z _p on the Spin^c-bundles and the Z _p-invariant moduli space of the solutions of the Seiberg–Witten equations for a Spin^c-structure X. When the Z _p action on the determinant bundle det L acts non-trivially on the restriction L| over the fixed point set , we consider -twisted solutions of the Seiberg-Witten equations over a Spin^c-structure ' on the quotient manifold X/Z _p X', (0,1). We relate the Z _p -invariant moduli space for the Spin^c-structure on X and the -twisted moduli space for the Spin^c-structure on X'. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the -twisted moduli space. When Z _p acts trivially on L|, we prove that there is a one-to-one correspondence between the Z _p -invariant moduli space M( ^Zp and the moduli space M (") where ' is a Spin^c-structure on X' associated to the quotient bundle L/Z _p X'. vskip0pt When p = 2, we apply the above constructions to a Kahler surface X with b ₂ ⁺ (X) > 3 and H ₂(X;Z) has no 2-torsion on which an anti-holomorphic involution acts with fixed point set , a Lagrangian surface with genus greater than 0 and []2H ₂(H ;Z). If _K ^X 2 > 0 or K _X ² = 0 and the genus g()> 1, we have a vanishing theorem for Seiberg–Witten invariant of the quotient manifold X'. When K _X ² = 0 and the genus g()= 1, if there is a Z ²-equivariant Spin^c-structure on X whose virtual dimension of the Seiberg–Witten moduli space is zero then there is a Spin^c-structure " on X' such that the Seiberg-Witten invariant is ±1.     

Faithful Group Actions on Rooted Trees Induced by Actions of Quotients

We develop some new techniques of constructing (finite state) actions on rooted homogeneous trees and apply them to various groups. In particular we show that there is a faithful action of each amalgameted free product of the form ??_?? on a rooted homogeneous tree of finite degree, described by finite state automorphisms.     

A Compactification of the Space of Expanding Maps on the Circle

We show the space of expanding Blaschke products on S¹ is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure preserving topological covering maps of S¹. Research supported in part by the NSF.     

Thermodynamic Formalism and Singular Invariant Measures for Critical Circle Maps

As is well known, the renormalization group transformation in the space of analytic circle homeomorphisms with one cubic critical point and rotation number equal to the golden section has a single fixed point T ₀. We construct the thermodynamic formalism for the critical map T ₀ and use it to calculate the Hölder indices for the singular invariant measure of T ₀.     

On Approximation of Groups, Group Actions, and Hopf Algebras

We give new examples and criteria in the theory of approximation of groups by finite groups. Bibliography: 17 titles.     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.