  We investigate the generalization of loop gravity's twisted geometries to a
q-deformed gauge group. In the standard undeformed case, loop gravity is a
formulation of general relativity as a diffeomorphism-invariant SU(2) gauge
theory. Its classical states are graphs provided with algebraic data. In
particular closure constraints at every node of the graph ensure their
interpretation as twisted geometries. Dual to each node, one has a polyhedron
embedded in flat space R^3. One then glues them allowing for both curvature and
torsion. It was recently conjectured that q-deforming the gauge group SU(2)
would allow to account for a non-vanishing cosmological constant Lambda, and in
particular that deforming the loop gravity phase space with real parameter q>0
would lead to a generalization of twisted geometries to a hyperbolic curvature.
Following this insight, we look for generalization of the closure constraints
to the hyperbolic case. In particular, we introduce two new closure constraints
for hyperbolic tetrahedra. One is compact and expressed in terms of normal
rotations (group elements in SU(2) associated to the triangles) and the second
is non-compact and expressed in terms of triangular matrices (group elements in
SB(2,C)). We show that these closure constraints both define a unique dual
tetrahedron (up to global translations on the three-dimensional one-sheet
hyperboloid) and are thus ultimately equivalent.
