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Register a new userRealizing compactly generated pseudo-groups of dimension one
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Authors
Gael Meigniez
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Many compactly generated pseudo-groups of local transformations on 1-manifolds are realizable as the transverse dynamic of a foliation of codimension 1 on a compact manifold of dimension 3 or 4.
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We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a primitive L-shaped Veech surface $X$ ramified over the singularity and a non-periodic connection point $Pin X$ has such a Veech group. Hubert and Schmidt showed that these Veech groups are infinitely generated and of the first kind. We use a result of Roblin and Tapie which connects the critical exponent of the Veech group of the covering with the Cheeger constant of the Schreier graph of $mathrm{SL}(X)/mathrm{Stab}_{mathrm{SL}(X)}(P)$. The main task is to show that the Cheeger constant is strictly positive, i.e. the graph is non-amenable. In this context, we introduce a measure of the complexity of connection points that helps to simplify the graph to a forest for which non-amenability can be seen easily.
We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Denote by $DC(M)_0$ the identity component of the group of compactly supported $C^infty$ diffeomorphisms of a connected $C^infty$ manifold $M$, and by $HR$ the group of the homeomorphisms of $R$. We show that if $M$ is a closed manifold which fibers over $S^m$ ($mgeq 2$), then any homomorphism from $DC(M)_0$ to $HR$ is trivial.
Let $S$ be a compact orientable surface, and $Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b in Mod(S)$ are independent (i.e., $[a^n,b^m] ot=1$ for any $n,m ot=0$) pseudo-Anosov elements. Then for any $n,m ge M$, the subgroup $<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $<a^n,b^m>$ is free of rank two and convex-cocompact if $|n|+|m| ge N$ and $nm ot=0$.
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