Commutator length of symplectomorphisms

By Michael Entov. 2001

48 citations recorded by the source corpus.

Abstract Each element $x$ of the commutator subgroup $[G, G]$ of a group $G$ can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of $x$. The commutator length of $G$ is defined as the supremum of commutator lengths of elements of $[G, G]$. We show that for certain closed symplectic manifolds $(M,\omega)$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian symplectomorphisms of $(M,\omega)$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega)$. By a different method we also show that in the case $c_1 (M) = 0$ the group $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover ${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity component of the group of symplectomorphisms of $(M,\omega)$ have infinite commutator length.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e035347575a2f0a9ff6f2d2e2e7c165840cd0fc3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Calabi quasimorphism and quantum homology (2002, background): “This construction was later generalized in [13] to closed symplectic manifolds (M,ω) with c1(M) = 0 (e.”
Calabi quasi-morphisms for some non-monotone symplectic manifolds (2005, background): “xistence of homogeneous quasi-morphisms on the group of Hamiltonian diffeomorphisms and/or its universal cover is known for some classes of closed symplectic manifolds (see e.g. Barge–Ghys [4], Entov [9], Gambaudo–Ghys [13] and Givental[14]). Inarecent work[11], EntovandPolterovichshowed–by usingFlo”
Conjugation-invariant norms on groups of geometric origin (2007, background): “The commutator length on Ham(M,ω) is known to be stably unbounded for various closed symplectic manifolds (see [3, 10, 11, 13, 22, 23]) including all surfaces and complex projective spaces of arbitrary dimension.”
Quasi-states and symplectic intersections (2004, background): “[19], [20]) yields that for a certain E ∈ R, depending on φ, c(f φ) = c(f )+ E, c(f φ−1) = c(f )− E.”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “uality has numerous generalizations (to other Lie groups, and to homeomorphisms of the circle [34]). The proof used here applies to linear symplectic groups. One can consider results such as those in [5, 16], proved using pseudo-holomorphic sections of symplectic brations over surfaces, as nonlinear”
On Continuity of Quasimorphisms for Symplectic Maps (2012, unknown): “Recently, several authors discovered that certain groups of diffeomorphisms preserving a volume or a symplectic form carry homogeneous quasimorphisms; see [5, 7, 17–19, 22, 41, 44, 45].”