Timeline for Examples of infinitely presented non-LEF groups
Current License: CC BY-SA 4.0
8 events
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| Feb 23, 2021 at 12:39 | comment | added | YCor | Yes, sorry, I mean "non-injective on $S$". (Alternatively instead of $S$, I might require the existence of $s_0\in S-\{1_P\}$ such that every homomorphism from $P$ to any finite group maps $s_0$ to $1$, and existence of a homomorphism $P\to G$ not mapping $s_0$ to $1_G$). | |
| Feb 23, 2021 at 12:18 | comment | added | frafour | @YCor in your previous comment, do you mean "every homomorphism from $P$ to any finite group is non-injective on $S$"? | |
| Feb 23, 2021 at 10:33 | comment | added | YCor | I don't know a reference (it's quite a restatement of the definition) but I can post a proof in an answer. | |
| Feb 23, 2021 at 10:18 | answer | added | markvs | timeline score: 1 | |
| Feb 23, 2021 at 9:22 | comment | added | frafour | Thank you for the comment @YCor ! Could you give a reference for the last statement? | |
| Feb 23, 2021 at 9:15 | comment | added | YCor | Anyway, these non-residually finite f.p. subgroups are artifacts. What's needed is the following: a group $G$ is non-LEF iff there is a finitely presented group $P$, a finite subset $S\subset P$ such that every homomorphism from $P$ to any finite group is non-injective, and a homomorphism $P\to G$ that is injective on $S$. | |
| Feb 23, 2021 at 9:11 | comment | added | YCor | Each group with a non-LEF subgroup is non-LEF, which makes immediate to produce infinitely presented ones (just take direct product of lamplighter with a finitely presented non-RF group). Examples with no LEF quotient (you probably mean no nontrivial LEF quotient): there's a result saying that every f.g. group embeds in a simple f.g. group, and the proof always produces infinitely presented groups. The last question is too vague to have a definite answer. | |
| Feb 23, 2021 at 8:56 | history | asked | frafour | CC BY-SA 4.0 |