24 problems

Smillie–Vogtmann's conjecture. The rational Euler characteristic $\chi(\operatorname{Out}(F_n))$ is always negative, and its absolute value grows faster than exponentially as a fun…

For each $k\geq 1$, let $\mu_k\in H_{4k}(\mathrm{Out}\,F_{2k+2};\mathbb{Q})$ be the homology classes introduced through the trace maps. Morita's conjecture. These classes are all t…

Let $F_n$ be the free group of rank $n$, let $K$ be the compact group appearing in the paper, and let $\operatorname{Hom}(F_n,K)/K$ denote the quotient of the representation variet…

Let $F_n$ be the free group of rank $n$, and let $\operatorname{Out}F_n$ be its outer automorphism group. Stable cohomology conjecture. The stable rational cohomology of…

For each integer $i\geq1$, let … be the homology class constructed from Kontsevich's graph-homology construction. Morita's conjecture. The classes $\mu_i$ are non-trivial for all…

Let $T$ be the subgroup of $\mathrm{IA}_3$ generated by the triple-indexed Magnus generators $K_{123}$, $K_{231}$, and $K_{312}$, and let $N$ be the normal closure in…

Let $A$ be a connected large-type Artin group: that is, an Artin group defined by a connected simplicial graph whose edge coefficients satisfy $m_{ab}\geq 3$. A rigid chunk is a ch…

Let $\Gamma$ be a colorable HHG. A quasitree of metric spaces and a projection complex are associated to $\Gamma$ using the constructions of Bestvina–Bromberg–Fujiwara. Assume eith…

Hyperbolic graph conjecture. Although for $n\geq 4$ the graph ${\mathcal{P}\mathcal{G}}_n$ does not have the property that every exponentially growing automorphism acts on it with…

Virtual centralization or atoroidal free subgroup conjecture. Either $\mathcal{H}$ virtually centralizes $\varphi$, or $\mathcal{H}$ contains a subgroup isomorphic to $F_2$ in whic…

Let $D53D$ be a finitely generated free group, let $D52CD66D61\operatorname{Out}(D53D)$ act on the cyclic splitting complex $D9D$. An element is loxodromic if it acts loxodromicall…

Uniform power-subgroup conjecture. There is some uniform $m\geq 2$ such that one can take

Non-conjugacy and abundance conjecture. Most of these elements $\varphi_w$ are pairwise non-conjugate in $Out(F_3)$, and this method yields doubly exponentially many fully irreduci…

Let $F_n$ be a free group of rank $n$, let $\operatorname{Out}(F_n)$ be its outer automorphism group, and let $\mathcal{FF}_n$ be the free factor complex whose vertices are conjuga…

Handel–Mosher's conjecture. There exists a single $\varphi\in\mathcal{H}$ that is (fully) irreducible with respect to every multi-edge extension…

Integral homology conjecture. The map $\mathcal{G}^1_n(X)\to\mathcal{G}_n(X)$ induces an isomorphism on homology in degrees $2*\leq n-3$.

Let $n-m$ be odd, and let ${\mathrm{HGC}}_{m,n}^{r,h}$, with $r\geq 2$ and $h\geq 1$, be the hairy graph complex equipped with the decreasing filtration by defect and associated sp…

Let $S$ be a compact surface of genus $g\geq 2$ with one puncture. Let $\mathcal{AC}(S)$ be its arc-and-curve graph and let $\mathcal{FF}$ be the free factor graph. The natural map…

Infinite diameter conjecture. The complex $\mathcal{FS}(\Gamma;\mathcal A)$ has infinite diameter, and the action of $\operatorname{Out}(\Gamma;\mathcal A)$ on…

For each $k\geq 1$, the construction gives a class … corresponding under Kontsevich's isomorphism to the displayed primitive cohomology class. Non-triviality conjecture. These clas…

Redundant-representation conjecture. (a) The action of $\operatorname{Out}(F_n)$ on $R$ is ergodic.

Let $F_n$ be the free group of rank $n$, and let $\operatorname{Out}(F_n)$ denote its outer automorphism group. Bogopol'skii–Puga conjecture. For every positive integer $n$, there…

Canonical special-linear subgroup conjecture. There is a canonically embedded copy

Essential containment conjecture. A solvable subgroup of $\operatorname{Out} A_\Gamma$ that is not virtually nilpotent should be essentially contained in an embedded copy of…