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Register a new userThe realization and classification of topologically transitive group actions on $1$-manifolds
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Enhui Shi
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In this report, we first recall the Poincares classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys classification theorem for minimal orientation-preserving group actions on the circle. Then we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.
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Denote by $DC(M)_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^infty$ manifold $M$. We show that if $dim(M)geq2$ and $r eq dim(M)+1$, then any homomorphism from $DC(M)_0$ to ${Diff}^1(R)$ or ${Diff}^1(S^1)$ is trivial.
We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler implies that every action of a topological group $G$ on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Hellys selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.
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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.
In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of ${rm SU}(2),,{rm U}(2),,{rm SO}(3),,{rm SO}(3) times S^1$ and ${{rm Spin}}^{mathbb{C}}(3)$. We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism $f:L(p, -1)times S^1rightarrow L(p, -1)times S^1$, the induced isomorphism $(picirc fcirc i)_*$ maps each element in the fundamental group of $L(p, -1)$ to itself or its inverse, where $i:L(p,-1)rightarrow L(p, -1)times S^1$ is the natural inclusion and $pi:L(p, -1)times S^1rightarrow L(p, -1)$ is the projection.
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