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Here is a question I stumbled upon in my research.

Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many (abstract) commensurability classes?

Recall that two groups $H_1,H_2$ are abstractly commensurable if there exists finite-index subgroups $K_1\le H_1$ and $K_2\le H_2$ such that $K_1$ and $K_2$ are isomorphic.

One can suppose $G$ torsionfree (replace $G$ by a finite-index torsionfree subgroup). Then I believe we can rephrase the question as "Do the Lie subgroups with lattices of the Malcev completion of $G$ fall into finitely many isomorphism classes?". An idea is to prove that there are only finitely many isomorphism classes of simply connected nilpotent Lie groups in each dimension, but this seems to be wrong. (There are a few lists in small dimension, for instance the article A Cornucopia of Carnot groups in Low Dimensions by Le Donne and Tripaldi. This list contains infinite families, for instance (147E). Do these contain lattices?)

For the application I have in mind, a positive answer for nilpotent groups of class 2 is sufficient.

Here is a question I stumbled upon in my research.

Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many abstract commensurability classes?

Recall that two groups $H_1,H_2$ are abstractly commensurable if there exists finite-index subgroups $K_1\le H_1$ and $K_2\le H_2$ such that $K_1$ and $K_2$ are isomorphic.

One can suppose $G$ torsionfree (replace $G$ by a finite-index torsionfree subgroup). Then I believe we can rephrase the question as "Do the Lie subgroups with lattices of the Malcev completion of $G$ fall into finitely many isomorphism classes?". An idea is to prove that there are only finitely many isomorphism classes of simply connected nilpotent Lie groups in each dimension, but this seems to be wrong. (There are a few lists in small dimension, for instance the article A Cornucopia of Carnot groups in Low Dimensions by Le Donne and Tripaldi. This list contains infinite families, for instance (147E). Do these contain lattices?)

For the application I have in mind, a positive answer for nilpotent groups of class 2 is sufficient.

Here is a question I stumbled upon in my research.

Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many commensurability classes?

Recall that two groups $H_1,H_2$ are commensurable if there exists finite-index subgroups $K_1\le H_1$ and $K_2\le H_2$ such that $K_1$ and $K_2$ are isomorphic.

One can suppose $G$ torsionfree (replace $G$ by a finite-index torsionfree subgroup). Then I believe we can rephrase the question as "Do the Lie subgroups with lattices of the Malcev completion of $G$ fall into finitely many isomorphism classes?". An idea is to prove that there are only finitely many isomorphism classes of simply connected nilpotent Lie groups in each dimension, but this seems to be wrong. (There are a few lists in small dimension, for instance the article A Cornucopia of Carnot groups in Low Dimensions by Le Donne and Tripaldi. This list contains infinite families, for instance (147E). Do these contain lattices?)

For the application I have in mind, a positive answer for nilpotent groups of class 2 is sufficient.

Here is a question I stumbled upon in my research.

Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many (abstract) commensurability classes?

Recall that two groups $H_1,H_2$ are abstractly commensurable if there exists finite-index subgroups $K_1\le H_1$ and $K_2\le H_2$ such that $K_1$ and $K_2$ are isomorphic.

One can suppose $G$ torsionfree (replace $G$ by a finite-index torsionfree subgroup). Then I believe we can rephrase the question as "Do the Lie subgroups with lattices of the Malcev completion of $G$ fall into finitely many isomorphism classes?". An idea is to prove that there are only finitely many isomorphism classes of simply connected nilpotent Lie groups in each dimension, but this seems to be wrong. (There are a few lists in small dimension, for instance the article A Cornucopia of Carnot groups in Low Dimensions by Le Donne and Tripaldi. This list contains infinite families, for instance (147E). Do these contain lattices?)

For the application I have in mind, a positive answer for nilpotent groups of class 2 is sufficient.