2019/11/05 by Moreno, Víctor
#18G99 #20J06 #55R35 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1911.01893
This thesis concerns the study of the Bredon cohomological and geometric dimensions of a discrete group G with respect to a family \mathfrakF of subgroups of G. With that purpose, we focus on building finite-dimensional models for E_\mathfrakF ( G ). The cases of the family \mathfrakFin of finite subgroups of a group and the family \mathfrakVC of virtually cyclic subgroups of a group have been widely studied and many tools have been developed to relate the classifying spaces for \mathfrakVC with those for \mathfrakFin. Given a discrete group G and an ascending chain \mathfrakF0 ⊆ \mathfrakF1 ⊆ … ⊆ \mathfrakFn ⊆ … of families of subgroups of G, we provide a recursive methodology to build models for E_\mathfrakFr ( G ) and give certain conditions under which the models obtained are finite-dimensional. We provide upper bounds for both the Bredon cohomological and geometric dimensions of G with respect to the families (\mathfrakFr)r∈ℕ utilising the classifying spaces obtained. We consider then the families \mathfrakHr of virtually polycyclic subgroups of Hirsch length less than or equal to r, for r∈ℕ. We apply the results obtained for chains of families of subgroups to the chain \mathfrakH0 ⊆ \mathfrakH1 ⊆ … for an arbitrary virtually polycyclic group G, proving that the corresponding Bredon dimensions are both bounded above by h(G) + r, where h(G) is the Hirsch length of G. Finally, we give similar results for the same chain of families of subgroups and an arbitrary locally virtually polycyclic group as the ambient group, obtaining in this case the upper bound h(G) + r + 1.