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Ali Taghavi
Ali Taghavi
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1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group?

2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metric. Can $G$ be embbeded isometrically in a locally connected Lie group with its left invariant metric?

Remark: We emphasis on the word   "Locally connectedconnected" because of the example $G\subset \mathbb{R}\setminus\{0 \}$ with $G=\{\pm e^n\mid n\in \mathbb{Z} \}$ with multiplication.

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group?

2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metric. Can $G$ be embbeded isometrically in a locally connected Lie group with its left invariant metric?

Remark: We emphasis on word  Locally connected because of the example $G\subset \mathbb{R}\setminus\{0 \}$ with $G=\{\pm e^n\mid n\in \mathbb{Z} \}$ with multiplication.

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group?

2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metric. Can $G$ be embbeded isometrically in a locally connected Lie group with its left invariant metric?

Remark: We emphasis on the word "Locally connected" because of the example $G\subset \mathbb{R}\setminus\{0 \}$ with $G=\{\pm e^n\mid n\in \mathbb{Z} \}$ with multiplication.

Source Link
Ali Taghavi
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

Is every countable discrete group a subgroup of a non discrete Lie group?

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group?

2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metric. Can $G$ be embbeded isometrically in a locally connected Lie group with its left invariant metric?

Remark: We emphasis on word Locally connected because of the example $G\subset \mathbb{R}\setminus\{0 \}$ with $G=\{\pm e^n\mid n\in \mathbb{Z} \}$ with multiplication.