Timeline for Very transitive groups
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2020 at 20:19 | comment | added | YCor | @Carl-FredrikNybergBrodda how could an action on a tree (by tree automorphisms) be highly transitive? it should preserve the distance, which for a 2-transitive action implies the distance of all distinct pairs is the same. | |
| Nov 14, 2020 at 12:50 | comment | added | Carl-Fredrik Nyberg Brodda | @YCor Well, picking apart the history of defining a group abstractly (or indeed at all) is non-trivial, but in any case, as I mentioned, I'm aware of this usage. OP's phrasing made it seem like they possibly weren't, so I wanted to make it clear. | |
| Nov 14, 2020 at 12:44 | comment | added | YCor | No, you won't find any list because it is hopeless to classify such groups. In the literature there are, however, some results giving some information about the class of groups that admit a highly transitive (=$k$-transitive for every $k$) faithful action on an infinite countable set. The keyword "highly transitive" might be helpful to find references. | |
| Nov 14, 2020 at 12:42 | history | edited | YCor | CC BY-SA 4.0 |
formatting, changed tag
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| Nov 14, 2020 at 12:41 | comment | added | YCor | @Carl-FredrikNybergBrodda 50 years before being defined as abstract objects, groups were defined as groups of permutations of a set. This language hasn't completely disappeared. | |
| Nov 14, 2020 at 10:07 | comment | added | Carl-Fredrik Nyberg Brodda | It is not a group that is highly transitive. It is a group action that is. But sometimes one says that a group is highly transitive when it has a natural action. Free groups of at most countable rank admit an action which is highly transitive. Burger and Mozes constructed a natural action of certain 'universal groups' on regular trees in 2000, which they prove is highly transitive. I think you'll have a hard time listing 'all' examples. | |
| Nov 14, 2020 at 9:47 | review | First posts | |||
| Nov 14, 2020 at 10:10 | |||||
| Nov 14, 2020 at 9:46 | history | asked | Florian Nuytens | CC BY-SA 4.0 |