Geodesicity of the big-Teichmuller space for general compact Z

Determine whether the big-Teichmuller space ((Z), d_Möb), defined as the space of Möbius-equivalence classes of antipodal functions on a compact metrizable space Z equipped with the metric induced by cross-ratios, is geodesic for arbitrary compact metrizable spaces Z that admit antipodal functions.

Background

For finite sets Z, the paper proves that the big-Teichmuller space ((Z), d_Möb) is geodesic and geodesically complete, and gives an explicit parametrization by an open simplex. The authors then discuss extending the definition of the big-Teichmuller space to general compact metrizable spaces Z that admit antipodal functions.

While this extension yields a metric space structure ((Z), d_Möb), the authors explicitly note that it is unclear whether the geodesic property persists beyond the finite case. Establishing geodesicity in this general setting would strengthen the geometric understanding of these deformation spaces and unify the finite and infinite cases.