Given a symplectic surface (Σ, ω) of genus g ≥ 4, we show that the free group with two generators embeds into every asymptotic cone of (Ham(Σ, ω), dH), where dH is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds. * This paper was the outcome of the authors' work in the computational symplectic topology graduate student team-based research program held Hofer's geometryLet (M, ω) be a symplectic manifold. Given a smooth function H :where H t (p) := H(t, p). Let ϕ t H : M → M be the flow of the ODEẋ(t) = X t (x(t)), making sufficient assumptions to ensure that the flow is globally defined on the time interval [0, 1] (for example, we could take M to be compact). Inside the group Symp(M, ω) = {φ ∈ Diff(M ) : φ * ω = ω} of symplectomorphisms we have the subgroup of Hamiltonian diffeomorphisms Ham(M, ω), which consists of the time-one maps ϕ 1 H : M → M of flows as above. The group Ham(M, ω) is equipped with a geometrically meaningful bi-invariant metric introduced by Hofer. The resulting metric group is an important object of study in symplectic geometry. For φ ∈ Ham(M, ω), we define the Hofer norm φ H = inf H 1 0

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We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds. 1 Introduction Let (M, ω) be an open symplectic 2n-dimensional manifold. Denote by Ham(M) the group of Hamiltonian diffeomorphisms generated by compactly supported Hamiltonian functions. Denote by Ham(M) the universal cover of Ham(M). We write elements of Ham(M) as [{f t } t∈[0,1] ], where {f t } t∈[0,1] is a smooth path of Hamiltonian diffeomorphisms with f 0 = Id, and [{f t } t∈[0,1] ] stands for the homotopy class of {f t } t∈[0,1] with fixed end points. In what follows we use the notation Vol(M) := M ω n . Introduce the Calabi homomorphism Cal : Ham(M) → R, as Cal([{f t } t∈[0,1] ]) = * Partially supported by European Research Council advanced grant 338809

We prove that on certain closed symplectic manifolds a C 1 -generic cyclic subgroup of the universal cover of the group of Hamiltonian diffeomorphisms is undistorted with respect to the Hofer metric.