arXiv 1202.35362012-02math.DS

带边界曲面的自同态的畸变

Distortion for diffeomorphisms of surfaces with boundary

Kiran Parkhe

备注：: 13 pages. arXiv admin note: text overlap with arXiv:math/0404532

摘要：

如果$G$是一个由生成元$\{g_1,..., g_s\}$生成的群，我们说一个无限阶元素$f \in G$是$G$的一个畸变元素，只要$\displaystyle \liminf_{n \to \infty} \frac{|f^n|}{n} = 0$，其中$|f^n|$是$f^n$相对于给定生成元的字长。设$S$是一个紧致可定向曲面，可能有边界，$\Diff(S)_0$表示$S$的$C^1$同构群的单位分量.我们的主要结果是：若$S$至少有亏格2，且$f$是$\Diff(S)_0$的某个生成子群中的畸变元，则对每个$f$，有$\supp(μ) \subseteq \Fix(f)$-不变Borel概率测度$μ$ .在一个小的附加假设下，在较低的属中也是如此。对于$μ$，$S$上的Borel概率测度，用$\Diff_μ(S)$表示保持$μ$的$C^1$同构群。我们的主要结果暗示了一大类高阶格不允许有无限象的$\Diff_μ(S)$同态。这些结果将Franks和Handel的结果推广到了有边界的曲面。

英文摘要：

If $G$ is a finitely generated group with generators $\{g_1,..., g_s\}$, we say an infinite-order element $f \in G$ is a distortion element of $G$ provided that $\displaystyle \liminf_{n \to \infty} \frac{|f^n|}{n} = 0$, where $|f^n|$ is the word length of $f^n$ with respect to the given generators. Let $S$ be a compact orientable surface, possibly with boundary, and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result is that if $S$ has genus at least two, and $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0$, then $\supp(μ) \subseteq \Fix(f)$ for every $f$-invariant Borel probability measure $μ$. Under a small additional hypothesis the same holds in lower genus. For $μ$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $μ$ by $\Diff_μ(S)$. Our main result implies that a large class of higher-rank lattices admit no homomorphisms to $\Diff_μ(S)$ with infinite image. These results generalize those of Franks and Handel to surfaces with boundary.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文