Timeline for Groups acting on trees
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2019 at 12:39 | comment | added | YCor | In the case of a tree and an action not of general type, either the action is already locally $T$-elliptic, or there is an invariant axis, or an equivariant Busemann function: in the latter two cases we obtain a "factor action" $E\simeq\mathbf{Z}$ as in my previous comment. I'm expecting a substitute for this, but maybe this is too optimistic. | |
| Mar 2, 2019 at 12:35 | comment | added | YCor | (by $X$-elliptic I meant locally $X$-elliptic) | |
| Mar 2, 2019 at 12:26 | comment | added | AGenevois | But I know that, if $G$ does not contain $\mathbb{F}_2$, then there exists a $G$-invariant collection of hyperplanes $\mathcal{J}$ such that the cubulation $Y$ of the wallspace $(X, \mathcal{J})$ isometrically embeds into a Euclidean cube complex. So you have a $G$-equivariant map $X \to Y$, but $Y$ is not necessarily Euclidean. I guess its isometry group should be (locally finite)-by-(free abelian). Also, the kernel of the action on $Y$ is not necessarily $X$-elliptic, but is (locally $X$-elliptic)-by-(free abelian). | |
| Mar 2, 2019 at 12:26 | comment | added | AGenevois | I would say no, but I have to think about it. Is $f$ supposed to preserve some structure? Does it send a cube to a cube (of possibly different dimension)? A remark: it is not reasonable to ask the kernel of the $G$-action on $E$ to be $X$-elliptic; locally $X$-elliptic should be better. | |
| Mar 2, 2019 at 11:38 | comment | added | YCor | Do you know if the theorem has a more geometric version of the kind: if a group $G$ acting on a finite-dimensional CCC $X$ has no $F_2$ subgroup acting properly on $X$, then there is a $G$-equivariant map $f:X\to E$ to a Euclidean CCC (a finite product of copies of $\mathbf{Z}$), such that the kernel of the $G$-action on $E$ is $X$-elliptic? (Since $\mathrm{Aut}(E)$ is f.g. virtually abelian, this implies the theorem.) | |
| Mar 2, 2019 at 11:03 | comment | added | AGenevois | I replaced "elliptic" with "$X$-elliptic" to avoid confusion. I also corrected the mistake in the corollary, thank you. | |
| Mar 2, 2019 at 10:59 | history | edited | AGenevois | CC BY-SA 4.0 |
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| Mar 2, 2019 at 10:34 | comment | added | YCor | Still there's a minor error in the corollary: it can happen that $G$ is already acting locally elliptically (horocyclic case). Side remark: in the corollary, no need to pass to a finite index subgroup, except precisely in the case $G$ preserves an axis, where one might have to pass to a subgroup of index 2. | |
| Mar 2, 2019 at 10:32 | comment | added | YCor | Ah, sorry, it's confusing (at least to me) that "locally elliptic", which I usually know as an intrinsic property of the group, is used here as a property of the action, especially since the conclusion is a mixture of intrinsic properties of $G$ (the homomorphism onto an abelian group is not related to the action, as stated here) and extrinsic (being locally elliptic). | |
| Mar 2, 2019 at 10:25 | comment | added | AGenevois | I am not assuming anything about the action. Why do you think the statements shoudn't be correct? | |
| Mar 2, 2019 at 9:58 | comment | added | YCor | You seem to assume (metric) properness at some point. Both the theorem and its corollary are false otherwise. Anyway I guess that the Caprace-Sageev theory is powerful enough to encompass non-proper actions. | |
| Mar 2, 2019 at 9:41 | history | answered | AGenevois | CC BY-SA 4.0 |