A T G Algebraic & Geometric Topology

We investigate cohomological properties of fundamental groups of strictly developable simple complexes of groups X . We obtain a polyhedral complex equivariantly homotopy equivalent to X of the lowest possible dimension. As applications, we obtain a simple formula for proper cohomological dimension of CAT .0/ groups whose actions admit a strict fundamental domain; for any building of type .W; S/ that admits a chamber transitive action by a discrete group, we give a realisation of the building of the lowest possible dimension equal to the virtual cohomological dimension of W ; under general assumptions, we confirm a folklore conjecture on the equality of Bredon geometric and cohomological dimensions in dimension one; finally, we give a new family of counterexamples to the strong form of Brown’s conjecture on the equality of virtual cohomological dimension and Bredon cohomological dimension for proper actions.

我们研究严格可展的群的单纯复形\(X\)的基本群的上同调性质。我们得到一个与\(X\)等变同伦等价且维数尽可能低的多面体复形。作为应用，我们得到了其作用具有严格基本域的\(CAT(0)\)群的恰当上同调维数的一个简单公式；对于任何允许离散群进行室传递作用的\((W,S)\)型建筑，我们给出了一个维数尽可能低且等于\(W\)的虚拟上同调维数的该建筑的实现；在一般假设下，我们证实了关于一维的布雷东几何维数和上同调维数相等的一个民间猜想；最后，我们给出了关于恰当作用的虚拟上同调维数和布雷东上同调维数相等的布朗强猜想的一系列新的反例。