25 problems

Simon’s finiteness conjecture. There are only finitely many knot groups $H$ for which there is an epimorphism $G\twoheadrightarrow H$.

A closed hyperbolic 3-manifold is a closed 3-manifold admitting a complete hyperbolic metric, and its fundamental group is denoted by $\pi_1(M)$. A finite index subgroup…

Let $G$ be a Seifert $3$-manifold group. Its single equation problem asks whether a single equation over $G$ has a solution. Theorem B establishes decidability when the base orbifo…

Let $G$ be the fundamental group of a $3$-manifold. A generalized torsion element is a nontrivial element whose finitely many conjugates have product equal to the identity, and $G$…

The generalized torsion criterion. In the class of 3-manifold groups, the existence of a generalized torsion element is equivalent to non-bi-orderability.

Homological detection conjecture. If

Let $M$ be a closed hyperbolic 3-manifold, and write $\pi_1(M)$ for its fundamental group. A group has a free non-abelian quotient if it admits a surjective homomorphism onto a non…

Waldhausen's conjecture. There is a finite cover $\tilde M\to M$ with $b_1(\tilde M)>0$.

Let $K$ be a nontrivial knot. A pseudo-meridian is a nontrivial element of the knot group that normally generates the knot group but is not an automorphic image of a meridian. Saku…

A 3-dimensional Poincaré duality group is a group satisfying Poincaré duality in dimension $3$, while a 3-manifold group is the fundamental group of a compact 3-manifold. Manifold-…

A 3-manifold group means the fundamental group of a compact 3-manifold, and a group is Kähler if it is the fundamental group of a compact Kähler manifold. Goldman–Donaldson–Rezniko…

Let $M$ be a Seifert $3$-manifold with base orbifold $B$, and let $G=\pi_1(M)$. Suppose that the orbifold Euler characteristic satisfies $\chi(B)<0$. Negative-characteristic decida…

A group is an $R^*$-group if it has no generalized torsion, and it is bi-orderable if it admits a strict total ordering invariant under both left and right multiplication. For a…

Let $G$ be the fundamental group of a $3$-manifold. A non-trivial element $g\in G$ is a generalized torsion element if some non-empty finite product of conjugates of $g$ equals the…

Let $M$ be a compact $3$-manifold with prime decomposition … where the prime factors are as in (1.1) and categories (i)–(iv): the $M_i$ are closed prime $3$-manifolds with infinite…

Simon’s finite-length conjecture. There is a number $N_G$ such that every epimorphism sequence with $n\geq N_G$ contains an isomorphism.

Let $G$ be the fundamental group of a $3$--manifold. A generalized torsion element is a non-trivial element whose non-empty finite product of conjugates is the identity, and $G$ is…

Generalized torsion conjecture. If $\pi_1(M)$ is not bi-orderable, then it contains a generalized torsion element.

Planar boundary conjecture. Every visual boundary of $G$ is planar, and furthermore, $G$ is virtually the fundamental group of a compact 3-manifold.

Let $M$ be an admissible $3$-manifold with first Betti number $b_1(M)\geq 1$. Residual local indicability conjecture. The fundamental group $\pi_1(M)$ is residually locally indicab…

Generalized-torsion conjecture. $G$ is bi-orderable if and only if $G$ has no generalized torsion element.

Let $M$ be the $3$-manifold under consideration, and let $\pi_1M$ denote its fundamental group. The group $\pi_1M$ acts freely and cocompactly on a CAT(0) cube complex if the actio…

Mapping torus convergence-group conjecture. For any p-A-like element $\phi$ of $G$, the mapping torus subgroup $G_\phi$ acts on $S^2$ as a convergence group.

Promotion of pseudo-fibering conjecture. There are three possibilities: $G$ is elementary; $G$ is a Fuchsian group; or $G$ is a closed hyperbolic $3$-manifold group.

Simple loop conjecture. If $\varphi$ is non-injective, then $\ker(\varphi)$ contains an element represented by a simple closed curve.