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Abstract. First year students learn that the Taylor expansion $Z_T$ carries functions into power series, and that it has some nice algebraic properties (e.g. multiplicativity, $Z_T(fg)=Z_T(f)Z_T(g)$). It is less well known that the same game can be played within arbitrary groups: there is a natural way to say "a Taylor expansion $Z$ for elements of an arbitrary group $G$", and a natural way to carry the algebraic properties of the Taylor expansion to this more general context. In the case of a general $G$ "Taylor expansions" (expansions with the same good properties as $Z_T$) may or may not exist, may or may not be unique, may or may not separate group elements, and a further good property which is hidden in the case of $Z_T$, "quadraticity", may or may not hold.
The purpose of this expository note is to properly define all the notions in the above paragraph, to enumerate some classes of groups whose theory of expansions we either understand or wish to understand, to indicate the relationship between these notions and the notions of "finite type invariants" and "unipotent" and "Mal'cev" completions, and to point out (with references) that our generalization of "expansions" to arbitrary groups is merely the tip of an iceberg, for almost everything we say can be generalized further to "expansions for arbitrary algebraic structures".
Archive figs ExQu3up.pdf ExQu.pdf ExQuTagged.pdf HelpNeeded-1602.pdf / abstract.txt recycling.txt
<< Notebook Pages >>
| Page | Created (UT) | Last Modified (UT) | |
|---|---|---|---|
| 1 | The co-product on A(G) | 2015-01-05 19:54:42 | 2015-01-07 14:47:02 |
| 2 | --- Double Filtrations | 2015-01-07 15:11:08 | 2015-01-07 15:23:07 |
| 3 | Gr(PBn) from generators and relations | 2015-01-25 19:27:57 | 2015-01-25 19:43:38 |
| 4 | Random | 2015-02-02 16:23:44 | 2015-02-02 16:26:16 |
| 5 | Products and Semi-Direct Products | 2015-04-17 16:38:28 | 2015-04-17 22:45:26 |
| 6 | Quillen's Theorem | 2015-05-08 18:28:24 | 2015-08-12 17:31:11 |
| 7 | --- Quillen's theorem - degree 2 testing | 2015-08-12 16:42:32 | 2015-08-17 14:02:42 |
| 8 | --- Quillen's theorem - degree 2 testing, 2 | 2015-08-12 16:42:32 | 2015-08-17 14:40:43 |
| 9 | The LCS expansion | 2015-06-22 15:34:51 | 2015-08-12 13:59:39 |
| 10 | scratch 150902 | 2015-09-02 16:00:31 | 2015-09-02 16:27:21 |
| 11 | Scratch 150907 | 2015-09-07 15:58:20 | 2015-09-07 16:19:15 |
| 12 | Degree by degree constructions | 2015-10-14 13:36:38 | 2015-10-14 14:06:31 |
<< Mathematica Notebooks >>
| Notebook (.pdf) | Source (.nb) | Created | Last Modified | Summary | |
|---|---|---|---|---|---|
| 1 | index | source | 2024-03-17 11:28:51 | 2016-02-05 10:34:32 | |
| 2 | ProgressiveScanning | source | 2024-03-17 11:28:51 | 2016-05-12 18:53:14 |
<< Links >>
- Studied 2015-02-01 12:36:58: Groupprops.
- Studied 2015-02-01 12:57:24: braid groups - Almost-direct product and 1-formality - MathOverflow.
- Studied 2015-02-01 13:13:57: Goetz-Conner_Talk.
- Studied 2015-02-23 15:12:24: [0705.4201v1] Bardakov, Bellingeri- On residual properties of pure braid groups of closed surfaces.
- Studied 2015-02-23 15:13:38: [math0512155v2] Bellingeri, Garvais, Guaschi- Lower central series for surface braid groups.
- Studied 2015-02-23 15:16:25: [1001.4471v2] Bellingeri, Gervais- Surface framed braids.
- Studied 2015-02-23 18:25:55: [1302.6536] Guaschi, Pineda- A survey of surface braid groups and the lower algebraic K-theory of their group rings.
- Studied 2015-10-09 11:52:14: Expositiones Mathematicae - Journal - Elsevier.
- Studied 2016-04-10 13:27:39: hopf algebras - The augmentation filtration on a group ring - MathOverflow.
- Studied 2016-05-18 17:25:28: The LaTeX Font Catalogue - Augie.
- Studied 2017-07-02 22:01:56: [1703.03448] Margalit, Winarski- The Birman-Hilden theory.
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