A Dehn function for Sp(2n, ℤ)

Abstract

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least 3 should have at most polynomial Dehn function. We prove that the lattice Sp(2p; Z) has quadratic Dehn function when p ≥ 5. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted. Gromov's conjectureOur main theorem verifies a special case of a conjecture of Gromov about lattices in symmetric spaces. We now explain what this conjecture is, why it is widely believed, and how previous authors have attacked it. First though, we must recall some basic definitions about Dehn functions (for Riemannian manifolds), symmetric spaces, lattices, and horoballs.Dehn functions for Riemannian Manifolds. Let X be a complete, simply connected Riemmanian manifold. Then given a piecewise smooth loop γ : S 1 → X, we can find a piecewise smooth function f : D 2 → X whose restriction ∂f to the boundary is γ. The area of γ is defined to be the smallest area of any such filling, i.e., Area(γ) = inf{Area(f )|∂f = γ} 2 Main TheoremsWe will presently reduce our main theorem to Theorems 2.2, 2.1, and 2.3, which will be proved in other sections. To state these theorems, we first need to recall the definition of the standard maximal parabolic subgroups of the symplectic group along with some notions related to filling. Symplectic linear algebraHere we briefly recall the theory of symplectic linear algebra. (An inspired reference is [19]).Definition. The symplectic form ω is the bilinear form on R 2p given by ω(v, w) = v T J 0 w • e α (x)e α (y) = e α (x + y).

A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

“…M g,1 ). From the result of Cohen [2], it follows that the distortion of I g,0 (resp. I g,1 ) in M g,0 (resp.…”

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

“…Moreover, Rafi-Schleimer [18] proved that an orbifold covering map of orientable surfaces induces a quasi-isometric embedding between the mapping class groups. For examples of distorted subgroups of mapping class groups, see Broaddus-Farb-Putman [4], Cohen [5], and Kuno-Omori [12], where it is proved that the Torelli group « b g is distorted in Mod.S b g /. Moreover, it has been proved by that the handlebody group is exponentially distorted in the mapping class group of the boundary surface.…”

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

“…For examples of distorted subgroups of mapping class groups, Broaddus-Farb-Putman [4], Cohen [5], and Omori and the second author [13] proved that the Torelli group I b g is distorted in Mod(S b g ). Moreover the handlebody group is exponentially distorted in the corresponding mapping class group by Hamenstädt-Hensel [12].…”