| Summary: | In the first part of the thesis we discuss the rank conjecture of Benson and Carlson. In particular, we prove that if G is a finite p-group of rank 3 and with p odd, or if G is a central extension of abelian p-groups, then there is a free finite G-CW-complex homotopy equivalent to the product of rk(G) spheres; where rk(G) is the rank of G.
We also treat an extension of the rank conjecture to groups of finite virtual cohomological dimension. In this context, for p a fixed odd prime, we show that there is an infinite group L satisfying the two following properties: every finite subgroup G<L is a p-group with rk(G)<3 and for every finite dimensional L-CW-complex homotopy equivalent to a sphere, there is at least one isotropy subgroup H<L with rk(H)=2.
In the second part of the thesis we discuss the study of homotopy G-spheres up to Borel equivalence. In particular, we provide a new approach to the construction of finite homotopy G-spheres up to Borel equivalence, and we apply it to give some new examples for some semi-direct products.
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