166 problems

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…

Bounded-region conjecture. The feature vectors of weights of the Whitehead graphs of elements from $F$ are separated into bounded regions in the corresponding space. Each such regi…

Let $F$ be a free group, and associate to each element of $F$ the feature vector of weights of its Whitehead Graph in the corresponding feature space. A region is a bounded subset…

Let $F$ be a free group. For each positive integer $n$, let $U_n$ be the set of all non-minimal elements in $F$ of length $n$, and let $NU_n\subset U_n$ be the subset consisting of…

Let $M$ be a topological $4$-manifold, let ${\vee}^3 S^1$ denote the wedge of three circles, and let ${\mathcal S}^0(Wh(Bor))$ denote the zero-framed surgery on the Whitehead doubl…

Let $n\geq 2$, let $\phi_1,\ldots,\phi_n$ be a collection of elements of $SU(2)$, and let $L^2_0(SU(2))$ denote the orthogonal complement of the constant functions in $L^2(SU(2))$.…

Free-group ergodicity conjecture. If $r\geq 3$, the action of $\mathsf{Out}(\pi)$ on $\mathsf{Hom}(\pi,G)$ is ergodic.

Let $F_r$ be a free group, let $NMin\subset F_r$ be the set of non-minimal elements, and let $WC(w)$ denote Whitehead complexity. Polynomial Whitehead-descent time-complexity conje…

Let $F_r$ be a free group, and let $T=T_r$ be the restricted set of Whitehead automorphisms described in the source. For each length $l$, let $NMin_l\subset F_r$ be the set of non-…

Exponential word-length growth conjecture. There exists a constant $c_r>1$ such that

Let $F=F_n$ be a free group of rank $n$, and let $NMin_l\subset F$ be the set of all non-minimal elements of length $l$. Let $LR(w)$ denote the length-reducing complexity associate…

Let $F_r$ be a free group of rank $r\geq 2$, and interpret “almost all” using asymptotic density on the set of elements of $F_r$ ordered by word length. An element is Whitehead min…

Let $F_r$ be a free group of rank $r$, let $F_r^k$ denote the set of $k$-tuples of elements of $F_r$, and let $U_{\min}$ be a minimal representative of the orbit of $U$ under the r…

Let $F_n$ be the free group on $n>1$ generators. Write $LO(F_n)$ for the space of left-orderings of $F_n$ and $BiO(F_n)$ for the space of bi-orderings, each with the topology inher…

Kazhdan property conjecture. For $k\geqslant 4$, $\operatorname{Aut} F_k$ satisfies Kazhdan property $(T)$.

Let $r\in\mathbb{N}$, let $\alpha\in\mathbb{C}[\mathbb{F}_r]\otimes\mathbb{C}[\mathbb{F}_r]$, and let $u$ be an $r$-tuple of free Haar unitaries. For almost every sequence of Haar-…

Conjecture on the non-prime density. The logarithmic density satisfies

Let $n\ge 3$, let $F_n$ be a free group of rank $n$, and let $F$ be a free group. Let $g,h:F_n\to F$ be monomorphisms. Sharp equalizer rank conjecture. Then … The paper constructs…

Infinite equalizer conjecture. There is an upper bound, depending only on $n$, for the number of $\operatorname{Eq}(\phi,\psi)$-orbits of the regular points of…

Stable compressed-rank conjecture. For every Stallings graph $\Gamma$ and $d\in\mathbb{N}$,

Let $G_{\alpha}=\left\langle \begin{bmatrix}1&1\\0&1\end{bmatrix},\begin{bmatrix}1&0\alpha&1\end{bmatrix}\right\rangle<\operatorname{SL}_2(\mathbb{C})$, and call a parameter non-f…

Let $G_{\alpha}=\left\langle \begin{bmatrix}1&1\\0&1\end{bmatrix},\begin{bmatrix}1&0\alpha&1\end{bmatrix}\right\rangle<\operatorname{SL}_2(\mathbb{C})$, and call $\alpha$ free whe…

Let $F_1$ and $F_2$ be finitely generated free groups, let $\varphi\colon F_1\to F_2$ be a homomorphism, and let $\tau^{(2)}_u(\varphi)$ denote its universal $L^2$-torsion. Univers…

Let $G$ be a finite group that is not Dedekind, let $R$ be the coefficient ring, and let $\operatorname{Bic}_R(G)$ be the group generated by all bicyclic units of $RG$. Fix a bicyc…

Let $F_2$ be the free group on two generators. A group is fully residually finite simple if, for every finite subset of its nontrivial elements, there is a homomorphism to a finite…