  We give new bounds on the cosystolic expansion constants of several families
of high dimensional expanders, and the known coboundary expansion constants of
order complexes of homogeneous geometric lattices, including the spherical
building of $SL_n(F_q)$. The improvement applies to the high dimensional
expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and
Oppenheim.
  Our new expansion constants do not depend on the degree of the complex nor on
its dimension, nor on the group of coefficients. This implies improved bounds
on Gromov's topological overlap constant, and on Dinur and Meshulam's cover
stability, which may have applications for agreement testing. In comparison,
existing bounds decay exponentially with the ambient dimension (for spherical
buildings) and in addition decay linearly with the degree (for all known
bounded-degree high dimensional expanders). Our results are based on several
new techniques:
  * We develop a new "color-restriction" technique which enables proving
dimension-free expansion by restricting a multi-partite complex to small random
subsets of its color classes.
  * We give a new "spectral" proof for Evra and Kaufman's local-to-global
theorem, deriving better bounds and getting rid of the dependence on the
degree. This theorem bounds the cosystolic expansion of a complex using
coboundary expansion and spectral expansion of the links.
  * We derive absolute bounds on the coboundary expansion of the spherical
building (and any order complex of a homogeneous geometric lattice) by
constructing a novel family of very short cones.
