14 problems

A limit group is a finitely generated fully residually free group. Let $F^{\mathbb{Z}[t]}$ be Lyndon's free exponential group. Myasnikov–Remeslennikov embedding conjecture. Every l…

Let $\Gamma_1,\dots,\Gamma_n$ be non-abelian limit groups and let $S\subseteq \Gamma_1\times\dots\times\Gamma_n$ be a subdirect product that intersects each $\Gamma_i$ non-triviall…

Direct-product conjecture. $S$ has a subgroup of finite index that is itself a direct product of $n$ or fewer limit groups.

Let $G_1,\dots,G_n$ be limit groups and let $\Gamma\subset G_1\times\cdots\times G_n$ be a subgroup. The relevant stronger result replaces “elementarily free groups” by “limit grou…

Let $G$ be a limit group, also called a fully residually free group, and let a group act properly and cocompactly by isometries on a $\mathop{\rm CAT}(0)$ space with the Isolated F…

Characterization conjecture. The group $G$ is elementarily equivalent to a non-abelian free group if and only if $G$ is a non-elementary hyperbolic fully residually free tower.

A limit group is a finitely generated subgroup of a group obtained from a free group by finitely many extensions of centralizers, and it is one-ended when it has one end. For a gro…

Let $G$ be a locally compact group. A limit group means a group in the sense of Sela, and $F_2$ denotes the free group on two generators. Dense limit-group conjecture. If $G$ conta…

Let $\zeta_{GR}(L)$ denote the ordinal of the well-ordered set of growth rates associated with a group $L$. Theorem 3.2 growth ordinal conjecture. The conclusion of Theorem 3.2 sho…

For a non-elementary hyperbolic group, let the growth ordinal be the ordinal of the well-ordered set of exponential growth rates over all finite generating sets. For a limit group…

Let $F$ be a free group and let $F^{\mathbb{Z}[t]}$ denote its exponential group. A finitely generated group is fully residually free if every finite set of its nontrivial elements…

Let $\Gamma_1,\dots,\Gamma_n$ be non-abelian limit groups, and let $P\leq\Gamma_1\times\dots\times\Gamma_n$ be a full subdirect product. The finiteness-property equivalence conject…

Let $G$ be a locally compact group, and let $F_2$ denote the free group of rank two. A group has a dense copy in $G$ if it contains a subgroup isomorphic to that group whose image…

Let $L_1,\ldots,L_n$ be non-abelian limit groups with $n\geq 2$, let $D=L_1\times\cdots\times L_n$, and let $H\leq D$ be a subdirect product that intersects each factor non-trivial…