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Register a new userG-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data
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Authors
Ralph Grieder
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Let $G$ be a finite group. To every smooth $G$-action on a compact, connected and oriented Riemann surface we can associate its data of singular orbits. The set of such data becomes an Abelian group $B_G$ under the $G$-equivariant connected sum. The map which sends $G$ to $B_G$ is functorial and carries many features of the representation theory of finite groups. In this paper we will give a complete computation of the group $B_G$ for any finite group $G$. There is a surjection from the $G$-equivariant cobordism group of surface diffeomorphisms $Omega_G$ to $B_G$. We will prove that the kernel of this surjection is isomorphic to $H_2(G;Z)$. Thus $Omega_G$ is an Abelian group extension of $B_G$ by $H_2(G;Z)$. Finally we will prove that the group $B_G$ contains only elements of order two if and only if every complex character of $G$ has values in $R$. This property shows a strong relationship between the functor $B$ and the representation theory of finite groups.
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We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.
Let $BS(1,n) =< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ ngeq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $. This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid. We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $<f,h | h circ f circ h^{-1} = f^n>$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $TT^2$. When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.
Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of the representation theory of finite groups and thus describes a geometric representation theory. We will prove that B_G consists only of copies of Z and Z/2Z. Furthermore we will show that there is a surjection from the G-equivariant cobordism group of surface diffeomorphisms to B_G. We will define a G-signature which is related to the G-signature of Atiyah and Singer and prove that this new G-signature is injective on the copies of Z in B_G.
In this work, we investigate the dynamical and geometric properties of weak solenoids, as part of the development of a calculus of group chains associated to Cantor minimal actions. The study of the properties of group chains was initiated in the works of McCord 1965 and Fokkink and Oversteegen 2002, to study the problem of determining which weak solenoids are homogeneous continua. We develop an alternative condition for the homogeneity in terms of the Ellis semigroup of the action, then investigate the relationship between non-homogeneity of a weak solenoid and its discriminant invariant, which we introduce in this work. A key part of our study is the construction of new examples that illustrate various subtle properties of group chains that correspond to geometric properties of non-homogeneous weak solenoids.
Let $G$ be a compact connected Lie group with $pi_1(G)congmathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.
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