Timeline for Sequence of epimorphisms of residually finite groups stabilizes
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25 events
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| Feb 21, 2021 at 12:41 | comment | added | ARG | @AGenevois ah yes.. that would be a good way to go at it. Actually I messed up: $C_2 \wr (C_2 \wr \mathbb{Z})$ is not Liouville, so its speed is linear and $\alpha \leq 1/2$. So as long as $H_2$ as a higher exponent (which seems very likely), this is a third possibility to exclude embeddings in $H_2$. | |
| Feb 20, 2021 at 18:59 | comment | added | AGenevois | @ARG: I have to check the details, but $H_2$ seem to act on a finite-dimensional CAT(0) cube complex with quasi-isometrically embedded orbits, so its (equivariant) Hilbert space compression should be 1. | |
| Feb 20, 2021 at 18:02 | comment | added | ARG | @Ycor & AGenevois: in a different direction, it could be also sufficient to compare the compression of these groups. But I don't know if the compression of either groups ( $C_2 \wr (C_2 \wr \mathbb{Z})$ or $H_2$) is known. The lower bounds I can gather are definitively not enough: $\alpha\big(C_2 \wr (C_2 \wr \mathbb{Z}) \big) \geq 4/7$ and $\alpha(H_2) \geq 1/2$. | |
| Feb 20, 2021 at 12:21 | comment | added | YCor | One big difference between $C_2\wr Z$ and $H_2$ is that in $H_2$ I expect there are permutations supported by $\{-n,\dots,n\}$ that have size $\gg n$ (quadratic or even cubic, I'm not sure). Say, the permutation $m\mapsto -m$ of $\{-n,\dots,n\}$. This is rather analogous to $C_2\wr Z^2$, where there are configurations of quadratic size that are "supported" in the $n$-ball (that is, turning on all lights in the $n$-ball). | |
| Feb 20, 2021 at 10:00 | comment | added | AGenevois | @YCor: I know, but the point that $\mathbb{Z}_2 \wr H$ has finite dimensional asymptotic cones iff $H$ has linear growth shows that there is something specific with the line, and $H_2$ and $\mathbb{Z}_2 \wr \mathbb{Z}$ are similar in many ways. For instance, $H_2$ has also a "lamplighter interpretation" (i.e. an arrow moving and acting on a colored line). But I agree that this is speculative, and I don't have any strategy to attack the problem right now. | |
| Feb 20, 2021 at 8:34 | comment | added | YCor | Anyway the idea that the asymptotic cones of $H_2$ are 1-dimensional sound speculative. For instance, I checked long ago that the asymptotic cones of $C_2\wr Z^2$ are $\infty$-dimensional (and not 2-dimensional as a naive analogy with $C_2\wr Z$ would predict). | |
| Feb 19, 2021 at 23:06 | comment | added | AGenevois | @YCor: Right, but I have a finer argument in mind. I would like to use a specific configuration in $\mathbb{Z}_2 \wr (\mathbb{Z}_2 \wr \mathbb{Z})$ implying the infinite dimension of asymptotic cones and to "rescale" its image in $H_2$ under a coarse embedding in order to deduce that the asymptotic cones of $H_2$ should also be infinite-dimensional. However, I didn't write all the details, so there is no guarantee it actually works... | |
| Feb 19, 2021 at 21:17 | comment | added | YCor | @AGenevois But the asymptotic cone behaves badly under coarse embeddings, including group embeddings (well, between f.g. groups they induce Lipschitz maps, but which can be non-injective or non-proper). So asymptotics cones will not be of much use. (Still, the question about dim of asymptotic cone of $H_2$, is intriguing, independently; this could be asked separately.) | |
| Feb 19, 2021 at 20:52 | comment | added | ARG | AGenevois: just a side[-meta-]note [in case you did not know already]: if you do not use @Ycor, Ycor will not be notified that you answered to him. The person who wrote the post [you in this case] is always notified of the comments, but you have to put an "arobase" before the name of any other user in order to notify an answer. | |
| Feb 19, 2021 at 20:45 | comment | added | ARG | [at]AGenevois oh, that's also a nice idea! but I'm also not sure how to deal with it. What is clear is that all these groups have asymptotic dimension 1, since there is a Hurewicz-type formula for asymptotic dimension [in Dranishnikov&Smith, Asymptotic Dimension of Discrete Groups]: when $1 \to N \to G \to Q \to 1$ then $asdim(G) \leq asdim(N) + asdim(Q)$. Since here $N$ is locally finite, it has $asdim =0$ (and obviously here $asdim(Q) = 1$. So $asdim(G)=1$ (the lower bound follows from containing $\mathbb{Z}$). But probably looking at the asymptotic cone would reveal finer distinctions. | |
| Feb 19, 2021 at 17:39 | comment | added | AGenevois | Interesting. I think the same conclusion can be obtained by a careful analysis of asymptotic cones. I know that the asymptotic cones of $\mathbb{Z}_2 \wr (\mathbb{Z}_2 \wr \mathbb{Z})$ have infinite topological dimension, and I guess that the asymptotic cones of $H_2$ are one-dimensional (by analogy with $\mathbb{Z}_2 \wr \mathbb{Z}$). However, I don't how to prove the latter assertion. | |
| Feb 19, 2021 at 6:00 | comment | added | ARG | @AGenevois I think this is correct, The $\ell^2$-isopermetric profile of Houghton's group has been computed by Saloff-Coste & Zheng (that of $\mathbb{Z}_2 \wr (\mathbb{Z}_2 \wr \mathbb{Z})$ is also known from Pittet & Saloff-Coste). But by Delabie,Koivisto,Le Maitre & Tessera this profile is monotonous under coarse embeddings. Hence there can be no coarse embedding of iterated lamplighter in $H_2$. | |
| Feb 18, 2021 at 12:10 | comment | added | AGenevois | What about iterated wreath products? Does $\mathbb{Z}_2 \wr (\mathbb{Z}_2 \wr \mathbb{Z})$ virtually embeds into $H_2$? | |
| Feb 18, 2021 at 12:05 | comment | added | AGenevois | You're right. Now I feel stupid... | |
| Feb 18, 2021 at 11:22 | comment | added | YCor | @AGenevois sorry I'm stupid. But still, an embedding is given by the transposition $(0,1)$ on the one hand, and the shift $\dots \mapsto -3\mapsto -2\mapsto -1\mapsto 2\mapsto 3\mapsto 4\mapsto\dots$, fixing $0$ and $1$. | |
| Feb 18, 2021 at 10:43 | comment | added | ARG | @Ycor If you have already have one in mind, I'd be grateful for a (solvable residually finite) such $G$ which virtually does not embed in $H_2$. | |
| Feb 18, 2021 at 9:45 | comment | added | ARG | @AGenevois Thanks! Actually I was more wondering if there is a quasi-isometric embedding of such a $G$ (and yes, with $L$ infinite) in $H_2$. But since q.i. are too messy I reduced to subgroup, which is (as you point out) now too obvious. But I guess the question that such a $G$ does not virtually embed in $H_2$ can still be tackled by the counting argument Ycor pointed out. | |
| Feb 18, 2021 at 8:06 | comment | added | AGenevois | @YCor, I am not sure to understand your example: your element of order two is not a finitely supported permutation, right? So how do you embed $\mathbb{Z}_2 \oplus \mathbb{Z}$ in $H_2$? | |
| Feb 18, 2021 at 7:41 | comment | added | YCor | @ARG the answer is yes by a counting argument: there are $2^{\aleph_0}$ such groups but only countably many embed into $H_2$ (still it's not hard to produce examples). | |
| Feb 18, 2021 at 7:39 | comment | added | YCor | @AGenevois the element of order 2 $(2n\leftrightarrow 2n+1)_{n\in\mathbf{Z}}$ and $n\mapsto n+2$ commute. | |
| Feb 17, 2021 at 21:14 | comment | added | AGenevois | I would say that $\mathbb{Z}_2 \oplus \mathbb{Z}$ (and so $\mathbb{Z}_2 \oplus(\mathbb{Z}_2\wr\mathbb{Z})$ if you really want an (infinite locally finite)-by-$\mathbb{Z}$ group) does not embed in $H_2$. | |
| Feb 17, 2021 at 20:55 | comment | added | ARG | @Ycor Perhaps a separate (or stupid) question: Assume $G$ is finitely generated and $1 \to L \to G \to \mathbb{Z} \to 1$ where $L$ is infinite locally finite. Are there such $G$ which are not isomorphic to a subgroup of $H_2$? Perhaps need to require that $G$ is solvable and/or residually finite for a positive answer? | |
| Jan 27, 2021 at 8:23 | comment | added | ARG | As a further remark, $H_2$ is isomorphic to $\textrm{Sym}_{\textrm{fin}}(\mathbb{Z}) \rtimes \mathbb{Z}$ where $\textrm{Sym}_{\textrm{fin}}(X)$ are the finitely supported permutations on the set $X$ and $\mathbb{Z}$ acts on $X = \mathbb{Z}$ by translation (the "infinite cyclic permutation"). And another interesting property: the Houghton groups $H_n$ have the property $FP_{n-1}$ but not $FP_n$. | |
| Jan 25, 2021 at 22:17 | comment | added | YCor | Just as a remark, the "second Houghton group" $H_2$ was discovered much before Houghton (as it's consider by B.H. Neumann in the 30's). The third Houghton group is a quite nontrivial analogue of $H_2$, and, quite surprisingly then, is finitely presented. Another remark is that this example is covered by my remark to dodd's post (it's an (infinite locally finite)-by-cyclic finitely generated group, so it's no surprise it's the same argument since it's the same in this generality, which is much broader than only locally finite Coxeter with action on the Coxeter graph). | |
| Jan 25, 2021 at 21:26 | history | answered | AGenevois | CC BY-SA 4.0 |