The Berry's phase and the black hole

In quantum mechanics a geometric phase, also called Berry phase, is a phase difference that a given physical system acquires during a cycle in which the system itself is under the action of an adiabatic process. This phase is linked to the geometric properties of the system itself (which is a simplification, but for our purposes there is no need to go into too much detail).
It was discovered independently by Shivaramakrishnan Pancharatnam in 1956⁽¹⁾, Hugh Christopher Longuet-Higgins⁽²⁾ in 1958 and subsequently generalized by Michael Berry⁽³⁾ in 1984. This phase, although geometric, has measurable physical effects, for example in an interference experiment. An example of a geometric phase is Foucault's pendulum.
The most famous version of this experiment, designed by Léon Foucault, dates back to 1851 when the French physicist, with the aim of showing the rotation of the Earth around its axis, suspended a ball of 28 kilograms of lead coated with brass over a surface of sand using a 67 meter cable hooked to the top of the dome of the Panthéon in Paris. The plane of the pendulum was observed to rotate clockwise at approximately 11.3 degrees per hour, completing a full circle in 31.8 hours. A more refined examination shows that after 24 hours there is a difference between the initial and final orientation of the trace left on Earth which is equal to

Probably not

A group velocity faster than $c$ does not mean that photons or neutrinos are moving faster thsn the speed of light.

This is the conclusion of Fast light, fast neutrinos? by Kevin Cahill⁽¹²⁾. He start his briefly analisys from some experimental observations of superluminal group velocity. In these experiments researchers measure a speed of light faster and slower than $c$ in vacuum. The first observation was occured in 1982⁽¹⁾, but an interesting collection of work in this subject is in Bigelow⁽⁷⁾ and Gehring⁽¹¹⁾. Experimentally when some pulses journey into a highly dispersive media occur some exotic effects. One of these is the observation of a negative group velocity, that coincides with a superluminal speed.
In Bigelow's and Gehring's works wasn't a really theoretical explenation. For example Bigelow proposed the following explaination:

(...) as the combination of different absorption cross sections and lifetimes for Cr³⁺ ions at either mirror or inversion sites within the BeAl₂O₄ crystal lattice. The superluminal wave propagation is produced by a narrow “antihole” [612 Hz half width at half maximum (HWHM)] in the absorption spectrum of Cr³⁺ ions at the mirror sites of the alexandrite crystal lattice, and the slow light originates from an even narrower hole (8.4 Hz) in the absorption spectrum of Cr³⁺ ions at the inversion sites.

They also considered

(...) the influence of ions both at the inversion sites and at the mirror sites. In addition, the absorption cross sections are assumed to be different at different wavelengths.

The arrows indicate the locations of ion sites that have mirror or inversion symmetry. On the right, the corresponding energy-level diagrams for Cr³⁺ ions at the different sites are shown.