Connectivity results for surface branched ideal triangulations

We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is en-hanced to be a [increment]-complex. Branched triangulations are considered up to the b-transit equivalence generated by b-flips (i.e. branched diagonal exchanges) and isotopy keeping V pointwise fixed. We extend a well-known connectivity result for 'naked' triangulations; in particular, in the generic case when x(s) <0, we show that each branched triangula-tion is connected to any other if x(s) is even, while this holds also for odd (S) possibly after the complete inversion of one of the two branch-ings. Natural distribution of the b-flips in sub-families gives rise to re-stricted transit equivalences with nontrivial (even infinite) quotient sets. We analyze them in terms of certain structures of geometric/topological nature carried by each branched triangulation, invariant for the given restricted equivalence.