13 problems

Infinite Motion Conjecture for permutation groups. If $G$ has infinite minimal degree and finite subdegrees, then $G$ admits an asymmetric 2-coloring.

Branch-group H-property conjecture. Many of these properties do not hold for groups from the other two classes.

Let $G$ be a group such that the classifying space for proper $G$-actions, $\underline{E}G$, has a finite model. Let $A_{\operatorname{ho}}(G)=\pi_G^0(\underline{E}G)$, let…

Let $G$ be a group and let $F$ be a field of characteristic zero. Define … where the colimit uses induction along inclusions of finite subgroups. Covariant representation-ring conj…

Let $G$ be an infinite group, and suppose that independent random-walk samples satisfy the law $x^2=1$ with positive probability. Leemann–de la Salle conjecture. If $G$ satisfies…

Antolín, Martino and Ventura's meta-conjecture. For any “reasonable” sequence $M=(\mu_n)_{n=1}^\infty$ of probability measures on $G$, we have $\operatorname{dc}_M(G)>0$ if and onl…

Protasov–Slobodianiuk conjecture. Every infinite group $G$ of cardinality $\varkappa$ has a partition as above such that

Let $G$ be an infinite group of cardinality $\varkappa$, and let $\omega$ denote the set of nonnegative integers. A subset $S\subseteq G$ is left $\varkappa$-large if there is a su…

Strong orbit-equivalence rigidity conjecture. If $G$ and $H$ are strongly orbit-equivalent, then

Let $G$ be a 2-primitive permutation group on an infinite set $X$, meaning that $G$ is 2-transitive and its point stabilisers are primitive, and suppose that $G$ is not 3-homogeneo…

Orbit-equivalent groups conjecture. $G$ and $H$ belong to the list above. This would strengthen the main theorem by removing its assumption that the groups are not 2-transitive. Th…

Let $GL_\infty(q)=\bigcup_{n\geq 1}GL_n(q)$ be the group of finitary infinite invertible matrices over the finite field $\mathbb F_q$. For $g\in GL_\infty(q)$, let $c(g)$ be the ra…

Let $\mathbb{H}$ denote the Hawaiian earring group, and let $H$ be an uncountable homomorphic image of $\mathbb{H}$ that embeds in the fundamental group of a planar or one-dimensio…