Noncoherence of arithmetic hyperbolic lattices

Conjecture 1.1 is out of reach for nonarithmetic lattices in O.n; 1/ and SU.n; 1/, since we do not understand the structure of such lattices. However, all known constructions of nonarithmetic lattices lead to noncoherent groups: See the author, Potyagailo and Vinberg [28] for the case of Gromov–Piatetsky–Shapiro construction; the same argument proves noncoherence of nonarithmetic reflection lattices (see eg Vinberg [42]) and nonarithmetic lattices obtained via Agol’s construction [1]. In the case of lattices in PU.n; 1/, all known nonarithmetic groups are commensurable to the ones obtained via the construction of Deligne and Mostow [12]. Such lattices contain fundamental groups of complex-hyperbolic surfaces which fiber over hyperbolic Riemann surfaces. Noncoherence of such groups is proved by the author in [25]; see also Section 8.