Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: SL(2,Z)

Bottom of Page

1 to 5 of 5

- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJul 5th 2019
- PermaLink
Author: Urs
Format: MarkdownItexjust for completeness and ease of hyperlinking <a href="https://ncatlab.org/nlab/revision/SL%282%2CZ%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/SL%282%2CZ%29">current</a>

just for completeness and ease of hyperlinking

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeMar 13th 2025
- PermaLink
Author: Urs
Format: MarkdownItexbriefly added the statement ([here](https://ncatlab.org/nlab/show/SL(2%2CZ)#SandTgenerateSL2Z)) that $S, T \in SL_2(\mathbb{Z})$ generate that group <a href="https://ncatlab.org/nlab/revision/diff/SL%282%2CZ%29/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/SL%282%2CZ%29/5">v5</a>, <a href="https://ncatlab.org/nlab/show/SL%282%2CZ%29">current</a>

briefly added the statement (here) that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi><mo>∈</mo><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S, T \in SL_2(\mathbb{Z})</annotation></semantics></math>$ generate that group

diff, v5, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeMar 13th 2025
- PermaLink
Author: Urs
Format: MarkdownItexalso added statement of the presentation $$ SL_2(\mathbb{Z}) \;\simeq\; \big\langle S\,, T \,\big\vert\, S^4 = \mathrm{e} ,\, (S T)^3 = S^2 \big\rangle $$ <a href="https://ncatlab.org/nlab/revision/diff/SL%282%2CZ%29/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/SL%282%2CZ%29/6">v6</a>, <a href="https://ncatlab.org/nlab/show/SL%282%2CZ%29">current</a>

also added statement of the presentation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mi>S</mi><mspace width="0.16667em"/><mo>,</mo><mi>T</mi><mspace width="0.16667em"/><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="0.16667em"/><msup><mi>S</mi> <mn>4</mn></msup><mo>=</mo><mi mathvariant="normal">e</mi><mo>,</mo><mspace width="0.16667em"/><mo stretchy="false">(</mo><mi>S</mi><mi>T</mi><msup><mo stretchy="false">)</mo> <mn>3</mn></msup><mo>=</mo><msup><mi>S</mi> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">⟩</mo></mrow><annotation encoding="application/x-tex"> SL_2(\mathbb{Z}) \;\simeq\; \big\langle S\,, T \,\big\vert\, S^4 = \mathrm{e} ,\, (S T)^3 = S^2 \big\rangle </annotation></semantics></math>$
diff, v6, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeMar 13th 2025
- PermaLink
Author: Urs
Format: MarkdownItexmentioned also the alternative presentation $$ SL_2(\mathbb{Z}) \;\simeq\; \big\langle S, T \,\big\vert\, S^4 = \mathrm{e} ,\, (T S)^3 = \mathrm{e} ,\, S^2 (T S) = (T S)S^2 \big\rangle \,. $$ (This is attributed by [Blanc & Déserti 2012 §2.1](https://ncatlab.org/nlab/show/SL(2%2CZ)#BlancDéserti12) to [Newman 1972](https://ncatlab.org/nlab/show/SL(2%2CZ)#Newman72), but I have trouble extracting this exact statement from Newman's book.) <a href="https://ncatlab.org/nlab/revision/diff/SL%282%2CZ%29/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/SL%282%2CZ%29/7">v7</a>, <a href="https://ncatlab.org/nlab/show/SL%282%2CZ%29">current</a>

mentioned also the alternative presentation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mi>S</mi><mo>,</mo><mi>T</mi><mspace width="0.16667em"/><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="0.16667em"/><msup><mi>S</mi> <mn>4</mn></msup><mo>=</mo><mi mathvariant="normal">e</mi><mo>,</mo><mspace width="0.16667em"/><mo stretchy="false">(</mo><mi>T</mi><mi>S</mi><msup><mo stretchy="false">)</mo> <mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">e</mi><mo>,</mo><mspace width="0.16667em"/><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>T</mi><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>T</mi><mi>S</mi><mo stretchy="false">)</mo><msup><mi>S</mi> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> SL_2(\mathbb{Z}) \;\simeq\; \big\langle S, T \,\big\vert\, S^4 = \mathrm{e} ,\, (T S)^3 = \mathrm{e} ,\, S^2 (T S) = (T S)S^2 \big\rangle \,. </annotation></semantics></math>$
(This is attributed by Blanc & Déserti 2012 §2.1 to Newman 1972, but I have trouble extracting this exact statement from Newman’s book.)

diff, v7, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTime4 days ago
- (edited 4 days ago)
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * Filippo Callegaro, [[Fred Cohen]], Mario Salvetti: *The cohomology of the braid group $B_3$ and of $SL_2(\mathbb{Z})$ with coefficients in a geometric representation*, in: *Configuration Spaces*, CRM Series. Edizioni della Normale, Pisa (2012) [[arXiv:1204.5390](https://arxiv.org/abs/1204.5390), [doi:10.1007/978-88-7642-431-1_8](https://doi.org/10.1007/978-88-7642-431-1_8)] <a href="https://ncatlab.org/nlab/revision/diff/SL%282%2CZ%29/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/SL%282%2CZ%29/17">v17</a>, <a href="https://ncatlab.org/nlab/show/SL%282%2CZ%29">current</a>
added pointer to:
- Filippo Callegaro, Fred Cohen, Mario Salvetti: The cohomology of the braid group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>B</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">B_3</annotation></semantics></math>$ and of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_2(\mathbb{Z})</annotation></semantics></math>$ with coefficients in a geometric representation, in: Configuration Spaces, CRM Series. Edizioni della Normale, Pisa (2012) [arXiv:1204.5390, doi:10.1007/978-88-7642-431-1_8]
diff, v17, current

1 to 5 of 5

nForum

Discussion Feed

Not signed in

Site Tag Cloud

nLab > Latest Changes: SL(2,Z)