A Quillen stability criterion for bounded cohomology

We provide a version of Quillen's homological stability criterion for continuous bounded cohomology.In the companion paper we exploit this criterion to derive new bounded cohomological stability results for various families of classical groups. 20J06; 55T25, 57T101 Introductiongroups is called homologically stable if there exists a function r W N !N such that the respective inclusions induce isomorphisms H q .G r .q// H q .G r .q/C1/ H q .G r .q/C2/ in group homology for all q 2 N. 1 Any such function r is then called a stability range for the family.Homological stability (with a linear stability range) has been established for many families of classical linear algebraic groups, but also for various families of nonlinear groups of interest, such as mapping class groups or automorphism groups of free groups.A particularly successful method to establish homological stability, employed for example by Van der Kallen [14], Harer [12], Hatcher and Vogtmann [13], Essert [9], and Sprehn and Wahl [22], is based on Quillen's stability criterion, which can be stated as follows.Suppose that, for every r 2 N, we are given a -complex X.r / -also known as a semisimplicial set -endowed with a simplicial G r -action, and two natural numbers .r/ and .r/ with the following properties: (Q1) X.r / is .r/-acyclic, ie the reduced homology z H .X.r // vanishes up to degree .r/.(Q2) X.r / is .r/-transitive, ie there is only one G r -orbit of l-simplices for l 2 f0; : : : ; .r/g.(Q3) The complexes are .r/-compatible, ie the stabilizer of an l-simplex in X.r / is isomorphic to G r l 1 for every l < .r/, and these isomorphisms are compatible with inclusions of stabilizers in the sense of condition (MQ3a) below.Then .G r / r 0 is homologically stable provided minf .r/; .r/g ! 1 as r ! 1.In addition, a stability range can be computed explicitly from the functions and .See Quillen's unpublished notes [21], or the more recent treatments by Bestvina [1] and Sprehn and Wahl [22] for details.1 As usual in homological algebra, we enumerate starting from 0. Accordingly, we use the convention 0 2 N, and for any k 2 N, we denote by OEk the .kC1/-elementset f0; : : : ; kg N. We also set OE1 WD N.