343 problems

Let $H,K\leqslant \operatorname{Free}[A]$ be subgroups of a free group, and write $H^w=w^{-1}Hw$. The reduced rank $\operatorname{rrk}(H)$ is the maximum of…

Let $D$ be a division ring, and write $D^*$ for its multiplicative group of nonzero elements. Lichtman's conjecture. If $D$ is noncommutative, then $D^*$ contains a non-cyclic free…

Let $F_n$ be the free group of rank $n$, let $\mathrm{IA}_n=\mathcal{A}_n'(1)$ be the group of automorphisms of $F_n$ acting trivially on its abelianization, and let … be the lower…

Church–Farb representation stability conjecture. The sequence induced in cohomology and homology by this stabilization sequence satisfies representation stability.

Strengthened Hanna Neumann conjecture. The sum over conjugacy classes should satisfy

Let $F_n$ be a free group and let $\rho\colon F_n\to\operatorname{PSL}_2(\mathbb{C})$ be a discrete, faithful representation without parabolic elements. Minsky's primitive-stabilit…

Let $D$ be a division ring, and let $H$ be a subnormal subgroup of the multiplicative group $D^\times$ of $D$. Gonçalves–Mandel conjecture. The group $H$ contains a non-cyclic free…

Let $F_2$ be the rank-two free group, and consider a type-preserving representation $\rho:F_2\to\mathrm{PSL}(2,\mathbb{R})$, meaning a non-elementary representation for which the p…

Let an infinite word be aperiodic if it is not ultimately periodic, and let the palindromic length of a factor be the minimum number of palindromes whose product is that factor. Un…

Katada's conjecture. For every $i$ and every $n\geq 3i$, the representation $H^A_i(\operatorname{IA}_n,\mathbb{Q})$ is isomorphic to $W_i$.

Let $\mathbb{F}_2$ be the free group on two generators, and let $\mathrm{C}^*_r(\mathbb{F}_2)$ denote its reduced C-algebra. Kadison's conjecture. The algebra…

Freeness conjecture. Every maximal subgroup of $\operatorname{IG}(\mathcal{E})$ is a free group.

Let $(u,v)$ be a generator pair for $F_2$, and let $\rho\colon F_2\to SL(2,\mathbb C)$ be an irreducible representation satisfying the $BQ$-conditions: for every primitive element…

A group is non-amenable if it does not admit an invariant mean, and a non-abelian free subgroup is a free group on at least two generators. Von Neumann–Day conjecture. A group is n…

Fix $k\ge 2$ and $m\ge 1$, and let $F=F(a_1,\dots,a_k)$ be the free group of rank $k$. For an $m$-tuple $\sigma$ of elements of $F$, write $R(\sigma)$ for the corresponding set of…

Let $n\ge 2$. Let $F_n$ be a free group with rank $n$ and $F$ be a free group. Let $g,h:F_n\to F$ be monomorphisms. Stallings equalizer conjecture. Then … This conjecture was open…

A group has an automorphism tower formed by repeatedly taking its automorphism group. Baumslag's general conjecture proposed that a sufficiently symmetric group should have a very…

Direct-product profinite rigidity conjecture. If a finitely presented residually free group $\Gamma$ has the same finite quotients as $F\times F'$, then

Day–Putman's conjecture. The complex of partial bases $\mathcal{B}_n$ is $(n-2)$-connected.

Habiro–Katada conjecture. The isomorphism

Let $F_m$ be a free group of rank $m$, let $H<F_m$ be a subgroup, and let $\Gamma_H(z)$ denote its cogrowth series for $z\in\mathbb{C}$. The rational cogrowth series conjecture. Th…

Church–Farb invariant-cohomology conjecture. The $\operatorname{GL}(n,\mathbb{Z})$-invariant part of $H^*(\operatorname{IA}_n,\mathbb{Q})$ vanishes stably.

Let $F_2$ be the free group of rank two, and consider the power word problem allowing nested exponents. For a fixed constant nesting depth, write…

Parzanchevski–Puder free-extension conjecture.

Miasnikov–Ventura–Weil conjecture. For every $H\leqslant_{\mathrm{fg}} F_A$,