Distortion elements in group actions on surfaces

By J. Franks, M. Handel. 2004

77 citations recorded by the source corpus.

If $\G$ is a finitely generated group with generators $\{g_1,...,g_j\}$ then an infinite order element $f \in \G$ is a {\em distortion element} of $\G$ provided $\displaystyle{\liminf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result shows that if $S$ has genus at least two and if $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0$, then $\supp(\mu) \subset \Fix(f)$ for every $f$-invariant Borel probability measure $\mu$. Related results are proved for $S = T^2$ or $S^2$. For $\mu$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $\mu$ by $\Diff_{\mu}(S)$. We give several applications of our main result showing that certain groups, including a large class of higher rank lattices, admit no homomorphisms to $\Diff_{\mu}(S)$ with infinite image.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/db5ae56a97a5371c647e484d5b6725ae2fe0a111

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Citations in the corpus

Distortion in transformation groups (2005, background): “Sn/ , thought of as a discrete group.”
GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “Distorted elements naturally appear inside nilpotent groups, and have been extensively used to study rigidity phenomena of group actions on 2-manifolds [17, 45].”
Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T) (2016, result): “somorph ic to the Lie algebra of a group in Theorems 1.1, 1.3, or 1.4, then the conclusion of the corresponding theorem continues to hold for cocompact lattices in G. Combinedwith the main results of [FH] and [BRHW3] we obtainthe followingtheorem for actions of lattices on surfaces. Theorem 1.5 ([FH”