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A group $G$ is slender if every subgroup $H \leq G$ is finitely generated. This includes polycyclic-by-finite groups. Such groups are also called noetherian.

Suppose that $L$ is a $G$-limit group in the sense that $L$ has a finite generating set $X$ such that for every $n$, the ball $B_n$ of radius $n$ in the Cayley graph is isomorphic to some ball of radius $n$ in a Cayley graph of some subgroup of $G$. Then is it true that $L$ is itself slender? Does it change anything if $G$ is equationally noetherian?

What about smallness? If $G$ is small, then must a $G$-limit group also be small?

I suspect this may be a tough question, but maybe I'm missing something obvious. Any response would be much appreciated.

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  • $\begingroup$ Btw I don't know what you call "subspaces of marked subgroups of $G$". $\endgroup$ – YCor Nov 1 '16 at 21:22
  • $\begingroup$ I aggree that the original title was misleading. The subspace of marked subgroups of $G$ is the union of all markings of all subgroups $K\leq G$. $\endgroup$ – NWMT Nov 2 '16 at 12:35
  • $\begingroup$ OK, but I still don't understand the meaning of the question. In the space of marked groups on say $k$ generators, you have two subsets, say $W(G)$, the set of marked groups whose underlying group is isomorphic to a subgroup of $G$, and the subset $L$ of marked groups whose underlying group is free (on at least 2 generators???). (...) $\endgroup$ – YCor Nov 2 '16 at 22:52
  • $\begingroup$ (...) What do you mean by "Can $W(G)$ can be isolated from $L$?" If $L$ contains free groups on $\le 1$ generators, these are not disjoint. If you don't allow them, and if by "isolate..." you mean $W(G)$ and $L$ have disjoint closures, then it means you're asking whether the closure of $W(G)$ contains a non-abelian limit group, and since the set of group isomorphism classes in $\overline{W(G)}$ is stable under taking subgroups, it means asking whether $\overline{W(G)}$ contains a non-abelian free group. But I may misinterpret, since the question is quite vague. $\endgroup$ – YCor Nov 2 '16 at 22:55
  • $\begingroup$ What I actually wanted to ask was can a $G$-limit group contain a free subgroup. But I was awkwardly trying to formulate this in the language of the topology of marked spaces. I will delete that paragraph and pose the question independently as a followup. $\endgroup$ – NWMT Nov 3 '16 at 12:40
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1) Consider the semidirect product $G=\mathbf{Z}^2\rtimes\mathbf{Z}$ with action by the matrix $A=\begin{pmatrix}2 & 1 \\ 1 & 1\end{pmatrix}$. It has the presentation $$\langle t,x,y\mid txt^{-1}=x^2y,\; tyt^{-1}=xy,\; [x,y]=1\rangle.$$ (It is actually generated by $(t,x)$).

Claim: some limit of markings on $G$ fails to be noetherian (although $G$ is)

[Added: this result was proved by Luc Guyot (Limits of metabelian groups. Internat. J. Algebra Comput. 22 (2012), no. 4, 1250031; ArXiv link, MR link), showing (2nd item of Theorem A therein) that suitable markings accumulate to the free metabelian group on 2 generators.]

Proof of the claim: [Edit: It's enough for purposes here to prove something weaker than Guyot's theorem, namely showing that some limit admits $\mathbf{Z}\wr\mathbf{Z}$ as a subgroup.] Consider the generating subset $(t^{n^2+1},t^n,x)$. There is a unique homomorphism $j_n$ from the wreath product $\langle s,x\mid [s^mxs^{-m}, x]=1,\forall m \in \mathbb{Z}\rangle$ to $G$ mapping $s$ to $s:=t^n$ and $x$ to $x$. We claim that it is injective in a ball of radius $p_n$ going to infinity with $n$. Once this is proved, this shows that some limit of copies of $G$ (any accumulation point for this choice of marking) contains a copy of $\mathbf{Z}\wr\mathbf{Z}$ and hence is not noetherian.

To prove the claim: assume the contrary. This means that some nontrivial element $w$ of the wreath product such that $j_n(w)=1$ for infinitely many $n$ (say $n\in I$, infinite set of integers). Clearly $w$ should be in the kernel of the canonical map to $\mathbf{Z}$, and therefore can be written $w=P(x)$ with $P\in\mathbf{Z}[u^{\pm 1}]$ (Laurent polynomial ring). Here for $P=\sum a_nu^n$, $P(x)$ means $\prod s^nx^{a_n}s^{-n}$. (This is a standard identification, since $\mathbf{Z}\wr\mathbf{Z}\simeq\mathbf{Z}[u^{\pm 1}]\rtimes\mathbf{Z}$).

That $j_n(w)=1$ means that, in $\mathbf{Z}^2$, $P(A^n)x=0$. Since $A$ is irreducible, that $P(A^n)$ has nontrivial kernel implies that it's zero. So for all $n\in I$, we have $P(A^n)=0$. But the $A^n$ have distinct eigenvalues (except for $n$ and $-n$), so infinitely of them cannot be killed by a single polynomial, contradiction.

Note that $G$ is linear over $\mathbf{Z}$, hence equationally Noetherian.

2) Small groups. Consider, for instance, a f.g. group $H$ containing all finite groups as subgroups (e.g., Neumann's group $\mathrm{Sym}_0(\mathbf{Z})\rtimes\mathbf{Z}$, or its finitely presented analogue due to Houghton), with no free subgroup.

Claim: some limit of markings on $H$ contains a free subgroup (although $H$ doesn't)

Fix a sequence of finite marked groups $(V_n,x_n,y_n)$ tending to the free groups on 2 generators. Then consider a sequence of generating subset of $H$, as a concatenation of a fixed generating subset of $H$, and a pair $(x'_n,y'_n)$ in $H$ generating a subgroup isomorphic to $(V_n,x_n,y_n)$ as marked group. Then any accumulation point of this sequence of marked groups has the property that the last two generators freely generated a free group.

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    $\begingroup$ What is the pre-image of the generator $t^{n^2 + 1} \in G$ in $\mathbf{Z} \wr \mathbf{Z}$ under $j_n$? (I just can't help doing this side remark: $\mathbf{Z}[\lambda^{\pm 1}] \rtimes_{\lambda} \mathbf{Z}$ accumulates on the free metabelian group on $2$ generators whenever $\lambda \in \mathbf{C}$ is not a root of unity, see Thereom A of arxiv.org/abs/1007.1356.) $\endgroup$ – Luc Guyot Nov 1 '16 at 22:25
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    $\begingroup$ @LucGuyot $t^{n^2}+1$ is clearly not in the image of $j_n$. Here I don't describe the whole limit but I prove that any limit has a certain subgroup isomorphic to the wreath product $\mathbf{Z}\wr\mathbf{Z}$. The limit will be some quotient of $\Gamma=\mathbf{Z}\wr\mathbf{Z}^2$, maybe $\Gamma$ itself. $\endgroup$ – YCor Nov 1 '16 at 23:06
  • $\begingroup$ @Ycor. Thank you for that great answer! I have a follow up question, but I'm not sure what the correct MO etiquette is about doing this. $\endgroup$ – NWMT Nov 2 '16 at 12:39
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    $\begingroup$ Good etiquette depends on the followup question. If it's a question about the answer itself that can probably be addressed by a simple rewrite of the answer, then just put the question in a comment to the answer. But if the question is really beyond the scope of your original question and beyond the scope of the answer, one should just write a new MO question, with references to any relevant old question. Burying new questions in comments is, in general, not desirable. $\endgroup$ – Lee Mosher Nov 2 '16 at 15:07

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