Abstract
If is a finitely generated group with generators , then an infinite-order element is a distortion element of provided that where is the word length of in the generators. Let be a closed orientable surface, and let denote the identity component of the group of -diffeomorphisms of . Our main result shows that if has genus at least two and that if is a distortion element in some finitely generated subgroup of , then for every -invariant Borel probability measure . Related results are proved for or . For a Borel probability measure on , denote the group of -diffeomorphisms that preserve by . We give several applications of our main result, showing that certain groups, including a large class of higher-rank lattices, admit no homomorphisms to with infinite image
Citation
John Franks. Michael Handel. "Distortion elements in group actions on surfaces." Duke Math. J. 131 (3) 441 - 468, 15 February 2006. https://doi.org/10.1215/S0012-7094-06-13132-0
Information