# Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?

This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $$X$$ is a hyperbolic space. It's also vaguely related to this question.

Suppose that $$(X,d)$$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $$x_0,x_1 \in X$$ there exists an isometry $$f:X\rightarrow X$$ such that $$f(x_0)=x_1$$. Does there always exist a set $$Y\subseteq X$$ with the following properties?

• $$Y$$ is uniformly discrete, i.e. there is an $$\varepsilon > 0$$ such that for any distinct $$x,y\in Y$$, $$d(x,y) > \varepsilon$$.
• $$Y$$ uniformly compactly covers $$X$$, i.e. for some $$x \in Y$$ there is a compact set $$K \ni x$$ such that translates of $$K$$ under $$\{f \in \mathrm{Aut}(X) : f(Y) = Y\}$$ cover all of $$X$$.
• $$Y$$ is vertex-transitive, i.e. for any $$x,y\in Y$$ there is a isometry $$f:X \rightarrow X$$ such that $$y=f(x)$$ and $$f(Y)=Y$$.

EDIT: I realized I hadn't captured what I wanted with the second bullet point.

• Your assumptions are equivalent to that there exists a cocompact lattice in the isometry group of $X$. (The setwise-stabilizer of $Y$ in $Isom(X)$.) This is already false for some 3-dimensional homogeneous manifolds (solvable simply connected Lie groups). Oct 7 '19 at 0:24
• @Misha Do you have a reference for this? I'm having difficulty finding the right search phrases. Jun 11 '20 at 20:57
• For instance, check Thurston's book "Geometry of 3-manifolds." He describes 8 3-dimensional geometries which, by his definition, are the ones admitting an isometric discrete action with quotient of finite volume. But there is a 1-parameter family of 3-dimensional solvable Lie groups, mathoverflow.net/questions/165656/…. Jun 25 '20 at 0:24

The answer is no. A quick answer can be done as follows:

(1) Pansu proved (1989) that two Carnot Lie groups are quasi-isometric if and only if they are isomorphic.

(2) There exists continuum many non-isomorphic 7-dimensional Carnot Lie groups.

If $$Y$$ is a proper, uniformly discrete and isometry-transitive metric space, then $$Y$$ is QI to its isometry group $$G$$, which is locally compact. Assuming it QI to a nilpotent Lie group in addition implies (by results of Gromov/Losert/Trofimov) that $$G$$, modulo a compact normal subgroup, is discrete and virtually nilpotent. There are only countably many QI classes. Hence at least one of the examples in (2) yields (for every choice of left-invariant Riemannian metric) a negative answer to your question.

There is an alternative using hyperbolicity rather polynomial growth, consisting of a continuum family of negatively curved homogeneous 3-folds. That they're not QI is also a result of Pansu. That a totally locally compact group QI to it has to be compact-by-discrete is an immediate consequence of the 2-dimensional Hilbert-Smith conjecture (which is an old theorem).

(a) Reference: this is the "famous" 1989 Pansu paper: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French) Ann. of Math. (2) 129 (1989), no. 1, 1–60.

(b) The continuum family consists of the 3-step-nilpotent simply connected Lie groups $$G_t$$ with the Lie algebra with basis $$(e_i)_{1\le i\le 7}$$ and brackets $$12|4$$ (meaning $$[e_1,e_2]=e_4$$), $$13|5$$, $$23|6$$, $$16|7$$, $$25|t.7$$, $$34|(t-1).7$$; it can be found in most lists. For, say $$t$$ transcendental, it's not QI to any discrete group. Magnin checked that $$\frac{(t^2-t+1)^3}{t^2(t-1)^2}$$ is an isomorphism invariant.

(c) Gromov proved that a discrete group with polynomial growth is virtually nilpotent. Lozert extended the result to locally compact groups, showing in particular that these are compact-by-Lie (hence compact-by-discrete in the totally disconnected case). Reference: V. Losert. On the structure of groups with polynomial growth. Math. Z. 195 (1987) 109-117. For groups acting on graphs (the totally disconnected case can boil down to it) Trofimov, reached the same conclusion: V. Trofimov. Groups with polynomial growth. Math USSR Sb 51 (1985) 405-418.

(d) It follows from Pansu's computation of $$L^p$$-cohomology and QI-invariance of the latter, that the groups $$H_t=\mathbf{R}^2\rtimes\mathbf{R}$$, for $$t\ge 1$$, with action $$s\cdot (x,y)=(e^sx,e^{ts}y)$$, are pairwise non-quasi-isometric. Reference: P. Pansu. Cohomologie Lp en degré 1 des espaces homogènes. Potential Anal. 27 (2007), no. 2, 151-165. Both steps for these groups were reproved differently in: Y. Cornulier, R. Tessera. Contracting automorphisms and Lp-cohomology in degree one. Ark. Mat. 49 (2011), no. 2, 295–324.

(Actually, it can be shown that for $$t>1$$, the Lie group $$H_t$$ is not QI to any vertex-transitive graph of finite valency. The main analytic ingredient is also Pansu 1989 (Cor. 6.9), but not the "famous" 1989 Pansu paper. Namely; Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 2, 177-212. See the discussion in my "quasi-survey" https://arxiv.org/abs/1212.2229, §19.6.3)

(e) For the Hilbert-Smith conjecture in dimension 2, it is asserted with a proof in Montgomery-Zippin's book but reading the proof I found a quite serious mistake. But everything could be fixed using results in Kolev, Boris Sous-groupes compacts d'homéomorphismes de la sphère. Enseign. Math. (2) 52 (2006), no. 3-4, 193-214. Also the 3-dim Hilbert conjecture by Pardon fixes this, although in a much more complicated way.

• I have a question regarding the sentence"If $Y$ is proper, uniformly discrete...". Is this a fact about Carnot Lie groups, Lie groups in general, homogeneous manifolds in general, or locally compact homogeneous metric spaces in general? Oct 6 '19 at 19:53
• Changed "it's" to "$Y$". It's about arbitrary metric spaces.
– YCor
Oct 6 '19 at 20:55
• Do you have references for at least one of these arguments? I found the 1989 paper by Pansu but I haven't been able to track down some of the other elements. Jun 12 '20 at 0:14
• @JamesHanson I added a few references
– YCor
Jun 12 '20 at 7:42
• Thank you. I apologize if this is basic geometric group theory, but where can I find a reference for the fact that there are only countably many quasi-isomorphism classes of discrete virtually nilpotent groups? Jun 12 '20 at 17:31