16 problems

Let a group action on a metric space be locally elliptic if every cyclic subgroup has bounded orbits, and elliptic if the whole group has bounded orbits. Consider an action of a fi…

Maximal FAn subgroup conjecture. A subgroup $H\subset W$ is maximal $\operatorname{FA}_n$ if and only if

The FAn conjecture. The following are equivalent:

A supramenable group is a group $G$ such that every $G$-set $X$ supports an invariant finitely additive measure normalized on any given non-empty subset $A\subseteq X$. The fixed-p…

Property $(T_n)$ and property $(\operatorname{FA}_n)$ are higher analogues of Kazhdan's property $(T)$ and the fixed-point property for actions on suitable non-positively curved sp…

Let $A_\Gamma$ be an Artin group, let $C_\Gamma$ be its associated clique-cube complex, and let … be the action by left multiplication. For a subgroup $H\subset A_\Gamma$, write…

A group $G$ has property ${\rm F}\mathcal{C}'$ if every cellular action of $G$ on a finite-dimensional CAT$(0)$ cube complex is locally elliptic. A subgroup is non-trivial if it is…

A group $G$ has property ${\rm F}\mathcal{A}'$ if every simplicial action of $G$ on a tree without inversion is locally elliptic. A subgroup is non-trivial if it is not the trivial…

Fixed-point conjecture. Every locally elliptic action of a finitely generated group on a finitely dimensional nonpositively curved complex has a global fixed point. In particular,…

Locally elliptic actions conjecture. If the action is locally elliptic, then it is elliptic. In particular, if $G$ is a torsion group, then the action is elliptic.

Marquis's conjecture. The group $G$ has a global fixed point in $X\cup\partial X$.

A super-expander is a sequence of finite Cayley graphs whose expansion persists with respect to every uniformly convex Banach space, with suitable finite generating sets. Let…

Let $G$ be a locally compact second countable group. Its fixed-point spectrum on a Banach space is the set of $p\geq 1$ such that every action by affine isometries of $G$ on that s…

Let $S$ be a semisimple connected Lie group with at least two simple factors and no compact factor. An irreducible lattice in $S$ is a lattice satisfying the usual irreducibility c…

Let $S$ be a connected semisimple Lie group with no compact simple factor, whose Lie algebra has real rank at least $2$. An irreducible lattice is a lattice in $S$ satisfying the u…

Let $G$ be an almost connected locally compact group. Recall that a group has property $\operatorname{(FB)}$ when every measurable action of $G$ on a finite-rank building stabilise…