  We investigate structural aspects of JT gravity through its BF description.
In particular, we provide evidence that JT gravity should be thought of as (a
coset of) the noncompact subsemigroup SL$^+$(2,R) BF theory. We highlight
physical implications, including the famous sinh Plancherel measure. Exploiting
this perspective, we investigate JT gravity on more generic manifolds with
emphasis on the edge degrees of freedom on entangling surfaces and
factorization. It is found that the one-sided JT gravity degrees of freedom are
described not just by a Schwarzian on the asymptotic boundary, but also include
frozen SL$^+$(2,R) degrees of freedom on the horizon, identifiable as JT
gravity black hole states. Configurations with two asymptotic boundaries are
linked to 2d Liouville CFT on the torus surface.

High Energy Physics - Theory