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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexsome minimum, to connect to *[[Cayley graph]]* and *[[geometric group theory]]* <a href="https://ncatlab.org/nlab/revision/finitely+generated+group/1">v1</a>, <a href="https://ncatlab.org/nlab/show/finitely+generated+group">current</a>

   some minimum, to connect to Cayley graph and geometric group theory

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded ([here](https://ncatlab.org/nlab/show/finitely+generated+group#BraidGroupExample)) mentioning of the example of the braid group. <a href="https://ncatlab.org/nlab/revision/diff/finitely+generated+group/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/finitely+generated+group/5">v5</a>, <a href="https://ncatlab.org/nlab/show/finitely+generated+group">current</a>

   Added (here) mentioning of the example of the braid group.

   diff, v5, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 6 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$ <a href="https://ncatlab.org/nlab/revision/diff/finitely+generated+group/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/finitely+generated+group/5">v5</a>, <a href="https://ncatlab.org/nlab/show/finitely+generated+group">current</a>

   added pointer to:

   • Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 6 of: Symmetry (2021) $[$pdf$]$

   diff, v5, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am looking for a HoTT reference which would write out the construction of groups $G$ *generated* by * a set $S$ of generators, * subject to relations $R \;\subset\; Id_{\Sigma S}(pt,pt) \,\times\, Id_{\Sigma S}(pt,pt)$ as * the looping of * the 1-truncation of * the homotopy cofiber of * the evident function $\Big(\underset{(p_1, p_2) \colon R}{\sum} \, S^1\Big) \longrightarrow \Sigma S$ When I first scanned over Bezem et al.'s *[[Symmetry]]* a while back my impression was that something like that must be what they write down, but now that I have a closer look it seems that they don't quite get around to. (?) Does anyone else write out group generation using just 1-HITs -- in citable form?

   I am looking for a HoTT reference which would write out the construction of groups $G$ generated by

   • a set $S$ of generators,

   • subject to relations $R \;\subset\; Id_{\Sigma S}(pt,pt) \,\times\, Id_{\Sigma S}(pt,pt)$

   as

   • the looping of

   • the 1-truncation of

   • the homotopy cofiber of

   • the evident function $\Big(\underset{(p_1, p_2) \colon R}{\sum} \, S^1\Big) \longrightarrow \Sigma S$

   When I first scanned over Bezem et al.’s Symmetry a while back my impression was that something like that must be what they write down, but now that I have a closer look it seems that they don’t quite get around to. (?) Does anyone else write out group generation using just 1-HITs – in citable form?