Constructions ofE_𝒱𝒞andE_ℱℬ𝒞for groups acting on CAT(0) spaces

Abstract

If is a group acting properly by semisimple isometries on a proper CAT.0/ space X , then we build models for the classifying spaces E VC and E F BC under the additional assumption that the action of has a well-behaved collection of axes in X . We verify that the latter assumption is satisfied in two cases: (i) when X has isolated flats, and (ii) when X is a simply connected real analytic manifold of nonpositive sectional curvature. We conjecture that has a well-behaved collection of axes in the great majority of cases.Our classifying spaces are natural variations of the constructions due to Connolly, Fehrman and Hartglass [4] of E VC for crystallographic groups .18F25, 55N15; 20F65

On the finiteness of the classifying space for the family of virtually cyclic subgroups

“…[20]. Also Farley has given a construction of EG in [7] for some CAT(0) groups, however without controlling the dimension. Proof When v is of finite order, then |v| = 0.…”

Models for classifying spaces for $${\mathbb {Z}\rtimes \mathbb {Z}}$$ Z ⋊ Z

“…The constructions follow the work of W. Lück and M. Weiermann in [5], and of D. Farley in [1]. The latter uses the fact that Z ⋊ Z is a CAT (0) group as it acts by isometries on the plane, and the former follows a general construction.…”

Geometric dimension of groups for the family of virtually cyclic subgroups

“…Finite dimensional models for EG have been constructed for several interesting classes of groups, for example, word-hyperbolic groups (Juan-Pineda, Leary, [19]), relatively hyperbolic groups (Lafont, Ortiz, [21]), virtually polycyclic groups (Lück,Weiermann, [29]) and CAT(0)-groups (Farley,[9], Lück, [27]).…”