Reduced Bers boundaries of Teichmüller spaces

We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space lying there into a point.This boundary turns out to be independent of the basepoint, and the action of the mapping class group extends continuously to this boundary.This is an affirmative answer to Thurston’s conjecture.He also conjectured that this boundary is homeomorphic to the unmeasured lamination space by the correspondence coming from ending laminations.This part of the conjecture needs some correction: we show that the quotient topology of the reduced Bers boundary is different form the topology induced from the unmeasured lamination space.Furthermore, we show that every auto-homeomorphism on the reduced Bers boundary comes from a unique extended mapping class.We also give a way to determine the limit in the reduced Bers boundary for a given sequence in Teichmüller space.