Timeline for What is an example of a word hyperbolic group without a finite complete rewriting system?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| S Aug 17, 2013 at 21:12 | history | bounty ended | CommunityBot | ||
| S Aug 17, 2013 at 21:12 | history | notice removed | CommunityBot | ||
| S Aug 9, 2013 at 19:46 | history | bounty started | Victor | ||
| S Aug 9, 2013 at 19:46 | history | notice added | Victor | Draw attention | |
| Aug 7, 2013 at 9:59 | comment | added | Victor | A personal feel is that if you manage to find an example admitting (infinite) complete rewriting system, the LHS of the rules forming context-free non-regular language, then most definitely the group will not admit a finite complete system. Note that, even wrt to the same generating set though sometimes it's possible to find another infinite complete system with LHS forming a regular language | |
| Jul 15, 2013 at 19:29 | comment | added | Derek Holt | @Mark: Your Dehn algorithm example cannot be complete in general, because it is strictly length reducing. In the (silly) example $\langle x,y \mid x=y \rangle$, $x$ and $y$ would be irreducible words representing the same group element. But I don't even know if there is an example of a hyperbolic group with given generating set for which it is known that there is no finite complete rewriting system. There may be examples if we fix the generating set and insist that the rewriting system is length non-increasing. | |
| Jul 15, 2013 at 19:08 | comment | added | Benjamin Steinberg | It was that example in your book that prompted this question. | |
| Jul 15, 2013 at 19:05 | comment | added | user6976 | For surface groups, there is another complete rewriting system - by Susan Hermiller (see my book). Whether this one is confluent, I do not know, never thought about it. I think Susan or Derek Holt must know. | |
| Jul 15, 2013 at 19:02 | comment | added | Benjamin Steinberg | @MarkSapir, is such a rewriting system complete for surface groups? | |
| Jul 15, 2013 at 18:59 | comment | added | user6976 | By the way, every hyperbolic group has a complete rewriting system "at 1", i.e. a finite rewriting system which is terminating and confluent at 1: if $u=1$, and $u\to v, u\to w$, then $v\to 1, w\to 1$. Take all words of length $\le 100\delta$ that are equal to 1. For each such word $w$ include all relations $u\to v$ where $u$ is a subword of $w$, $|u|> 1/2|w|$, $v$ is the complement of $u$ in $w$ (considered as a cyclic word). Derek Holt probably knows examples when that rewriting system is not complete (in general). | |
| Jul 15, 2013 at 18:49 | comment | added | user6976 | I do not think there is an example. As a candidate, I would take the intermediate groups in the construction of a free Burnside group. | |
| Jul 15, 2013 at 18:36 | history | asked | Benjamin Steinberg | CC BY-SA 3.0 |