9 problems

Juan-Pineda–Leary conjecture. A classifying space ${\mathcal E}_{VC}(G)$ cannot be finite unless $G$ is virtually cyclic.

Let $G$ be the fundamental group of a closed hyperbolic manifold $M$ of dimension $d$. Write $obreakoperatorname{hdim}^G(underline{underline{E}}G)$ for the equivariant homotopy di…

Let $G$ be a finitely presented group satisfying the condition that there is a finite set of virtually cyclic subgroups $\{H_1,\ldots,H_n\}$ such that every virtually cyclic subgro…

Let $G$ be a group. It has BVC if there are virtually cyclic subgroups $V_1,\ldots,V_n$ such that every virtually cyclic subgroup of $G$ is conjugate to a subgroup of some $V_i$. B…

Let $G$ be a countable group, and let $\underline{\underline{\mathrm{cd}}}(G)$ and $\underline{\underline{\mathrm{gd}}}(G)$ denote its Bredon cohomological and geometric dimensions…

Finite-type classifying-space conjecture. The only groups admitting a finite type model for $\underline{\underline EG}$ are virtually cyclic.

Let $G$ be an elementary amenable group, and let $\uu EG$ denote a classifying space for the family of virtually cyclic subgroups. Bredon geometric-dimension conjecture. In light o…

Nonexistence conjecture. If $G$ is not virtually cyclic, then $\underline{\underline{E}}G$ has no finite model.

Let $G$ be a group. A finite model for $\operatorname E_{\mathcal{VC}}G$ is a finite-dimensional $G$-CW model with finitely many cells for the classifying space for virtually cycli…