arXiv subjects
Publications and source records attributed to Raeyong Kim.
We study the algebraic rank of various classes of $\mathrm{CAT}(0)$ groups. They include right-angled Coxeter groups, right-angled Artin groups, relatively hyperbolic groups and groups acting geometrically on $\mathrm{CAT}(0)$ spaces with isolated flats. As one of our corollaries, we obtain a new proof of a result on commensurability of Coxeter groups
We prove that every finite connected simplicial complex has the homology of the classifying space for some $\mathrm{CAT}(0)$ cubical duality group. More specifically, for any finite simplicial complex $X$, we construct a locally $\mathrm{CAT}(0)$ cubical complex $T_{X}$ and an acyclic map $t_{X} : T_{X} \to X$ such that $π_1(T_X)$ is a duality group.
We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.