24 problems

Uniform stable commutator length conjecture. For each $n$ there is a constant $C(n)>0$ such that, for every arithmetic hyperbolic $n$-manifold $M$ and every non-parabolic…

Let $F_S^{\mathbb{Q}}$ be the free $\mathbb{Q}$-group on a set $S$, and let $\operatorname{scl}$ denote stable commutator length. Rationality conjecture. For every element…

Spectrum conjecture. Both spectra coincide with the set of right-computable numbers.

Calegari's amenability conjecture. $G$ is amenable.

Let $G=F/\ncl{w}$ be a one-relator group and let $X$ be the associated presentation complex. A face-essential map is a map $f:Y\to X$ from a 2-complex $Y$ that is face-essential, a…

Uniform spectral gap conjecture. There is a uniform constant $B>0$ such that any hyperbolic $C$-special or $A$-special group has a spectral gap $B$ for integral chains.

Calegari's conjecture. For any integral chain $c$ in a free group,

Walker's conjecture. For any fixed chain $c$ in $F_n$, ${\rm scl}(c_{\bm{o}})$ is piecewise quasilinear in $1/o_i$: there are some $p\in\mathbb{Z}_+$ and a finite partition of…

Let $F=\langle a,b\rangle$ be a free group of rank $2$, and let $w\in[F,F]$ be a homologically trivial word. Regard $F$ as the fundamental group of the hyperbolic once-punctured to…

Let $M$ be a compact $3$-manifold, and let $G=\pi_1(M)$. A group is PQL if stable commutator length is piecewise rational linear on $B_1^H(G)$, the real vector space of one-boundar…

Let $G_j^k = \langle a_j \rangle / \langle a_j^{o_{j,k}} \rangle = \mathbb Z/o_{j,k}\mathbb Z$ and let $G_k = *_j G_j^k$. Fix $\Gamma$ such that $\Gamma \in B_1^H(G_k)$ for all $k$…

Let $M$ be a closed hyperbolic $d$-manifold. Fix some $\delta>0$. Let $\gamma$ be a random geodesic of length in $[n-\delta,n+\delta]$ conditioned to be homologically trivial, and…

Let $G$ be a hyperbolic group with finite generating set $S$, and let $\lambda$ be such that the number of elements of length $n$ is $\Theta(\lambda^n)$. Let $v$ be a random elemen…

Let $p,q,r\in\mathbb{N}$ and set $n=p+q+r$. Stable commutator length formula. Then … The paper presents this as an unrelated conjecture at its conclusion. No resolution or partial…

Let $F_2$ be the free group on two generators, and let $operatorname{scl}(F2)$ denote the set of stable commutator lengths of elements of $F2$. Rationality conjecture. The ima…

Interval conjecture. The set of values

Rank-two isometry conjecture. For every $g\in F_2$, one has

Let $G$ be a finitely presented group, and let $g_n$ be obtained by a length-$n$ random walk, conditioned to lie in $[G,G]$. Suppose that, for every $\epsilon>0$, there is a positi…

Let $G$ be any finitely generated group, let $S$ be a finite symmetric generating set, and suppose that the space $Q(G)$ of homogeneous quasimorphisms modulo homomorphisms is finit…

Let $(G,\mu,Y)$ be nondegenerate in the sense of Definition. Let $g_1,\ldots,g_k$ be obtained by independent even-length random walks on $G$ of lengths $\ell_1n,\ldots,\ell_kn$, co…

Let $(G,\mu,Y)$ be nondegenerate in the sense of Definition, and let $g$ be obtained by an even-length random walk on $G$, conditioned to lie in $[G,G]$. Concentration conjecture.…

Let $G$ be a hyperbolic group with finite generating set $S$, and let $\lambda$ be the growth entropy of $G$ with respect to $S$. For a random element $g$ of word length approximat…

For a nondegenerate random walk action $(G,\mu,Y)$, consider the unit ball of the stable commutator length norm on a random finite-dimensional subspace of $B_1^H(G)$. Cross-polytop…

Let $F_2$ be the free group on generators $a,b$, and let $w\in [F_2,F_2]$ be arbitrary. Let $S$ be a hyperbolic once-punctured torus with $\pi_1(S)\cong F_2$, and let…