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Qwert Otto
Qwert Otto
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The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal $I$) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, but without any proofs. If $G$ is further nilpotent, I could prove it using the fact that $G_n = (1+I^n)\cap G$ (for the notation and the proof, see this MO post and Theorem 12.1.6 of CDBooK by Chmutov-Duzhin-Mostovoy, respectively).

Any comments or references are appreciated.

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal $I$) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, but without any proofs. If $G$ is further nilpotent, I could prove it using the fact that $G_n = (1+I^n)\cap G$ (for the notation, see this MO post).

Any comments or references are appreciated.

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal $I$) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, but without any proofs. If $G$ is further nilpotent, I could prove it using the fact that $G_n = (1+I^n)\cap G$ (for the notation and the proof, see this MO post and Theorem 12.1.6 of CDBooK by Chmutov-Duzhin-Mostovoy, respectively).

Any comments or references are appreciated.

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Qwert Otto
Qwert Otto
  • 985
  • 2
  • 11

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal $I$) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, and in the first part of this MO post claims a nilpotent and torsion-freebut without any proofs. If $G$ is embedded intofurther nilpotent, I could prove it using the fact that $\hat G$ as a lattice$G_n = (1+I^n)\cap G$ (for the notation, but without proofs in either casesee this MO post).

Any comments or references are appreciated.

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, and in the first part of this MO post claims a nilpotent and torsion-free $G$ is embedded into $\hat G$ as a lattice, but without proofs in either case.

Any comments or references are appreciated.

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal $I$) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, but without any proofs. If $G$ is further nilpotent, I could prove it using the fact that $G_n = (1+I^n)\cap G$ (for the notation, see this MO post).

Any comments or references are appreciated.

Source Link
Qwert Otto
Qwert Otto
  • 985
  • 2
  • 11

Is Malcev completion an embedding?

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, and in the first part of this MO post claims a nilpotent and torsion-free $G$ is embedded into $\hat G$ as a lattice, but without proofs in either case.

Any comments or references are appreciated.