Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

Abstract

Vol. 0 nH 3 /. Then there must exist a sequence .g k / k2N of elements in PSL.n; C/ such that for every 2 0 , limwhere i W 0 ! PSL.2; C/ is the standard lattice embedding and n W PSL.2; C/ ! PSL.n; C/ is the irreducible representation.This phenomenon, called rigidity at infinity, was proved by the author and Francaviglia [14, Theorem 1.1] for n D 2 and any nonuniform lattice of PSL.2; C/ -notice that the same phenomenon holds for all rank-one representations of any rank-one lattice [21]. However, since in that case our proof exploited the existence of natural maps for nonelementary representations -see for instance Besson, Courtois and Gallot [2; 3; 4] and Francaviglia [13] -we could not use the same argument here.For our purposes, the existence of a boundary map ' k is crucial. Indeed, the possibility to express the Borel invariant ˇn. k / as the integral over a fundamental domain for 0 nPSL.2; C/ of the pullback of the Borel cocycle along the boundary map ' k together with the maximality hypothesis allows us to prove the existence of a suitable sequence .g k / k2N of elements in PSL.n; C/ such that the sequence .g k ' k . // k2N is bounded, where D .0; 1; e i=3 ; 1/ and is any element of 0 . The boundedness of the previous sequence implies the boundedness of .g k k g 1 k . // k2N for every 2 0 and hence we reach our conclusion.