Connectivity results for surface branched ideal triangulations

Abstract

We consider triangulations of closed surfaces S with a given set of vertices V ; every triangulation can be branched that is enhanced to be a Δ-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 χ(S) < 0, we show that each branched triangulation is connected to any other if χ(S) is even, while this holds also for odd χ(S) possibly after the complete inversion of one of the two branchings. Natural distribution of the b-flips in sub-families gives rise to restricted 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.