Extracting explicit conjugating paths for subsurface-deformation Hamiltonian flows

Determine an explicit procedure to compute, from a given G-invariant multi-function f: G^k → R and its variation function F, the boundary-conjugating paths g_i^t ∈ G (with g_i^0 = e) that cover the Hamiltonian flow of the induced function f_{\underline{\alpha}} on the character variety Hom(π1(S), G)//G in the case where the supporting subsurface S_0 is a fully separating m-holed sphere. Specifically, for each component S_i adjacent to S_0 along boundary curve c_i, derive g_i^t directly from f and F so that the flow Xi_ρ(t)(γ) = g_i^t ρ_i(γ) (g_i^t)^{-1} realizes the Hamiltonian dynamics on the restriction to Γ_i = π1(S_i).

Background

The paper shows that Hamiltonian flows of functions on surface-group character varieties induced by G-invariant multi-functions are subsurface deformations supported on appropriate subsurfaces. In the special case where the supporting subsurface S_0 is a fully separating m-holed sphere, the authors prove that the Hamiltonian flow can be covered by paths of conjugations along the boundary curves, i.e., by elements g_i^t ∈ G that conjugate the restricted representations on each complementary component S_i.

While this geometric description is established, the authors note that obtaining explicit expressions for these conjugating paths is nontrivial in practice. Although they compute some examples, they identify a gap: a systematic method to extract g_i^t from the data of the invariant multi-function f and its variation function F is currently lacking. Addressing this would enhance the practical computability of Hamiltonian flows and deepen understanding of their dynamics in concrete settings.