Countable subgroups of Euclidean space

Arnold W. Miller

April 2013

revised May 21, 2013

In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is K_{σ}, the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is K_{σ} (since it is countable) but not compactly generated (since compact subsets are finite).

Beros showed that for any Polish group G, every K_{σ} subgroup of G is compactly generated iff every countable subgroup of G is compactly generated. He showed that any countable subgroup of ℤ^{ω} (infinite product of the integers) is compactly generated and more generally, for any Polish group G, if every countable subgroup of G is finitely generated, then every countable subgroup of G^{ω} is compactly generated.

In unpublished work Beros asked the question of whether “finitely generated” may be replaced by “compactly generated” in this last result. He conjectured that the reals ℝ under addition might be an example such that every countable subgroup of ℝ is compactly generated but not every countable subgroup of ℝ^{ω} is compactly generated. We prove (Theorem 4) that this is not true. The general question remains open.

In the course of our proof we came up with some interesting countable subgroups. We show (Theorem 6) that there is a dense subgroup of the plane which meets every line in a discrete set. Furthermore for each n there is a dense subgroup of Euclidean space ℝ^{n} which meets every n − 1 dimensional subspace in a discrete set and a dense subgroup of ℝ^{ω} which meets every finite dimensional subspace of ℝ^{ω} in a discrete set.

Theorem 1 Every countable subgroup G of ℝ is compactly generated.

Proof If G has a smallest positive element, then this generates G. Otherwise let x_{n} ∈ G be positive and converge to zero. Let G = {g_{n} : n < ω}. For each n choose k_{n} ∈ ℤ so that |g_{n} − k_{n}x_{n}| ≤ x_{n}. Let

C = {0} ∪ {x_{n}, g_{n} − k_{n}x_{n} : n < ω}

then C is a sequence converging to zero, so it is compact. Also

g_{n} = (g_{n} − k_{n}x_{n}) + k_{n}x_{n}

so it generates G. QED

Theorem 2 For 0 < m < ω every countable subgroup G of ℝ^{m} is compactly generated.

Proof For any ϵ > 0 let

V_{ϵ} = span_{ℝ}({u ∈ G : ||u|| < ϵ})

where here the span is taken respect to the field ℝ. Note that for 0 < ϵ_{1} < ϵ_{2} that V_{ϵ_{1}} ⊆ V_{ϵ_{2}}. And since they are finite dimensional vector spaces there exists ϵ_{0} > 0 such that V_{ϵ} = V_{ϵ_{0}} whenever 0 < ϵ < ϵ_{0}. Let V_{0} = V_{ϵ_{0}}. Let ϵ_{0} > ϵ_{1} > ⋯ descend to zero. For each n let B_{n} ⊆ {u ∈ G : ||u|| < ϵ_{n}} be a basis for V_{ϵ_{n}} = V_{0}. Then |B_{n}| = dim(V_{0}) and the sequence (B_{n})_{n<ω} “converges” to the zero vector 0. Hence {0} ∪ ⋃_{n<ω} B_{n} is a compact subset of G. Let

G_{0} = span_{ℤ}( ⋃_{n<ω} B_{n})

where span_{ℤ}(X) is the set of all linear combinations from X with coefficients in ℤ or equivalently the group generated by X.

Claim. G_{0} is dense in V_{0}. Proof Let |B_{n}| = k_{0} = dim(V_{0}). Suppose v ∈ V_{0} and we are given ϵ_{n}. Then v = ∑_{u∈B_{n}} α_{u}u for some α_{n} ∈ ℝ. Choose n_{u} ∈ ℤ with n_{u} ≤ α_{u} < n_{u} + 1. Let w = ∑_{u∈B_{n}} n_{u}u_{n} Then

||v − w|| = || ∑_{u∈B_{n}}(α_{u} − n_{u})u|| ≤ ∑_{u∈B_{n}} |α_{u} − n_{u}| · ||u|| ≤ |B_{n}|ϵ_{n} = k_{0}ϵ_{n}

Since k_{0}ϵ_{n} → 0 as n → ∞ we are done. QED

Next we tackle the “discrete part” of G.

Claim. There exists a finite F ⊆ G such that G ⊆ span_{ℤ}(F) + V_{0}. Proof First note that for any u, v ∈ G, if ||u − v|| < ϵ_{0}, then u − v ∈ V_{0} and hence u + V_{0} = v + V_{0}. Now let F_{0} ⊆ G be finite so that

span_{ℝ}(F_{0}) = span_{ℝ}(G)

As we saw above for any u ∈ G there exists v ∈ span_{ℤ}(F_{0}) such that

||u − v|| ≤ N_{0} =^{def} ∑_{u∈F_{0}} ||u||

If no such F exists then we may construct an infinite sequence u_{n} ∈ G such that

u_{n} ∉ span_{ℤ}(F_{0} ∪ {u_{i} : i < n}) + V_{0} and ||u_{n}|| ≤ N_{0}.

But by compactness of the closed ball of radius N_{0} there would have to be i < j with ||u_{j} − u_{i}|| < ϵ_{0} and hence u_{j} ∈ u_{i} + V_{0} which is a contradiction. QED

Now let

C = {0} ∪ F ∪ ⋃_{n<ω} B_{n}

and note that C is a compact subset of G and span_{ℤ}(C) is dense in G. To finish we can repeat the argument for ℝ:

Let G = \{g_{n} : n < ω\}. Choose u_{n} ∈ span_{ℤ}(C) with ||g_{n} − u_{n}|| < \frac{1}{2^{n}}. Then C ∪ {g_{n} − u_{n} : n < ω} is compact and generates G. QED

Question 3 Do there exist Polish groups Γ_{1}, Γ_{2} such that both have the property that every countable subgroup is compactly generated but the same is not true for Γ_{1} × Γ_{2}?

Let S be the real line with the Sorgenfrey topology, i.e., half-open intervals [x, y) are basic open sets. Then (S, +) is not a topological group; addition is continuous but not subtraction. However countable subgroups are compactly generated since a descending sequence with its limit point is compact. However {(x, −x) : x ∈ S} is discrete in S × S, so the subgroup, so {(x, −x) : x ∈ ℚ} is not compactly generated.

Theorem 4 Every countable subgroup G ⊆ ℝ^{ω} is compactly generated.

Proof

Define ||x||_{N} = sup{|x(i)| : i < N}. So this is just the sup-norm but restricted to the first N-coordinates, i.e., ||x||_{N} = ||x ↾ N||. An open neighborhood basis for x is given by the sets {y : ||y − x||_{N} < ϵ} for N < ω and reals ϵ > 0. Similarly we put

G ↾ n =^{def} {u ↾ n : u ∈}.

For any n by applying the proof of Theorem 2 we may find ϵ_{n} > 0 such that if

V_{n} =^{def} span_{ℝ}({u ↾ n : u ∈ G and ||u||_{n} < ϵ_{n}}) ⊆ ℝ^{n}

then for any δ with 0 < δ ≤ ϵ_{n}

V_{n} = span_{ℝ}({u ↾ n : u ∈ G and ||u||_{n} < δ}).

Without loss we may assume that ϵ_{n} < \frac{1}{2^{n}}. Take B_{n} ⊆ G so that

|B_{n}| = dim(V_{n}) ≤ n and V_{n} = span_{ℝ}({u ↾ n : u ∈ B_{n}}).

By the argument of the first claim of Theorem 2, we have that for every u with u ↾ n ∈ V_{n} there exists v ∈ span_{ℤ}(B_{n}) with

||u − v||_{n} < n · ϵ_{n} < \frac{n}{2^{n}}

By the argument of the second claim there exists a finite F_{n} ⊆ G such that

G ↾ n ⊆ span_{ℤ}(F_{n} ↾ n) + V_{n}.

Hence for every u ∈ G there exists v ∈ span_{ℤ}(F_{n} ∪ B_{n}) with ||u − v||_{n} < \frac{n}{2^{n}}.

Apply this to each u ∈ F_{n+1}. So we may choose û ∈ span_{ℤ}(F_{n} ∪ B_{n}) with ||u − û||_{n} < \frac{n}{2^{n}}. Put

H_{n+1} =^{def} {(u − û) : u ∈ F_{n+1}}

and put

C = {0} ∪ ⋃_{n<ω}(B_{n} ∪ H_{n}).

Then C is compact, since for every n if u ∈ B_{n}, then we have that ||u||_{n} < \frac{1}{2^{n}} and if v ∈ H_{n+1}, then ||v||_{n} < \frac{n}{2^{n}}. By induction we have that F_{n} ⊆ span_{ℤ}(C) for each n, since if we assume F_{n} ⊆ span_{ℤ}(C), then for u ∈ F_{n+1} we have that û ∈ span_{ℤ}(C) and so u = (u − û) + û is in span_{ℤ}(C). It follows that span_{ℤ}(C) is dense in G since ⋃_{n<ω}(B_{n} ∪ F_{n}) is dense in G. By the argument used in Theorems 1 and 2 we get that G is compactly generated. QED

Question 5 For countable subgroups of ℝ^{ω} the generating compact set can always be taken to be a convergent sequence. We don’t know if this is more generally true.

The motivation for the following example was a doomed attempt to prove Theorem 2 for the plane by looking at one dimensional subspaces and considering multiple cases.

Theorem 6 There exists a dense subgroup G ⊆ ℝ^{2} such that for every line L in the plane, G ∩ L is discrete.

Proof Let M be a countable transitive model of a large finite fragment of ZFC. Working in M let ℙ be a countable family of nonempty open subsets of the plane which is a basis for the topology. Forcing with ℙ produces a generic point p ∈ ℝ^{2}.

The following facts are well-known.

1. If p is ℙ-generic over M then p is not in any closed nowhere dense subset of ℝ^{2} coded in M.

2. If (p, q) are ℙ^{2}-generic over M, then p is ℙ-generic over M and q is ℙ-generic over M[p].

3. If (p, q) are ℙ^{2}-generic over M, then p + q is ℙ-generic over M.

4. If p is ℙ-generic over M and q ∈ M ∩ ℝ^{2}, then p + q is ℙ-generic over M. And if α ∈ M ∩ ℝ is nonzero, then αp is ℙ-generic over M.

Now let ∑_{n<ω} ℙ be the countable direct sum of copies of ℙ. Let (p_{n} : n < ω) be ∑_{n<ω} ℙ-generic over M and let G = span_{ℤ}({p_{n} : n < ω}).

It is easy to see that the p_{i} are algebraically independent vectors over the field of rationals (or even the field M ∩ ℝ) so for any u ∈ G there is a unique finite F ⊆ ω such that let u = ∑_{i∈F} n_{i}p_{i} and each n_{i} ≠ 0 for i ∈ F. Let F = supp(u) (the support of u).

Suppose L is any line in the plane thru the origin and let

L^{+} = L \ {0}.

Note that L is determined by any element of L^{+}.

Claim. If u, v ∈ L^{+} ∩ G, then supp(u) = supp(v). Proof Let supp(u) = F_{0} and supp(v) = F_{1}. Suppose for contradiction that F_{1} \ F_{0} is nonempty. Let i ∈ F_{1} \ F_{0} and put F = F_{1} \ {i}. Then

v = k_{i}p_{i} + ∑_{j∈F} k_{j}p_{j}.

It follows from the well-known facts above that p_{i} is ℙ-generic over the model N =^{def} M[(p_{j} : j ∈ F_{0} ∪ F)]. Also since q =^{def} ∑_{j∈F} k_{j}p_{j} ∈ N and k_{i} ∈ ℤ we know that v = k_{i}p_{i} + q is ℙ-generic over N. On the other hand the line L is coded in N since u is in this model. But L is a closed nowhere dense subset of the plane so by genericity v ∉ L. QED

Let F_{0} be the common support of all u ∈ L^{+}∩G. Fix any i_{0} ∈ F_{0}. Choose u ∈ L^{+} ∩ G such and u = ∑_{i∈F_{0}} n_{i}p_{i} and n_{i_{0}} > 0 is the minimal possible positive coefficient of p_{i_{0}} and for any point in L^{+} ∩ G.

Claim. L ∩ G = {nu : n ∈ ℤ}. Proof Let v = ∑_{i∈F_{0}} m_{i}p_{i} be in L^{+}∩G. Then m_{i_{0}} must be divisible by n_{i_{0}} otherwise n_{i_{0}} would not have been minimal. Say n · n_{i_{0}} = m_{i_{0}} for some n ∈ ℤ. Then note that v − n · u is in L ∩ G and its support is a subset of F_{0} \ {i_{0}}. Hence v − n · u = 0 and we are done. QED

Now suppose that L is a line not containing 0. If L ∩ G is nonempty take u_{0} ∈ L ∩ G. Let L_{0} = −u_{0} + L. Since L_{0} is line thru the origin there exists u_{1} ∈ G such that G ∩ L_{0} = {nu_{1} : n ∈ ℤ}. Then

G ∩ L = {u_{0} + nu_{1} : n ∈ ℤ}.

QED

Remark 7 .

1. Note that dense-in-itself is not the same as dense, for example, ℤ × ℚ is dense-in-itself but not dense in ℝ^{2}.

2. Finitely generated does not imply discrete. The additive subgroup of ℝ generated by \{1, \sqrt{2}\} is dense.

3. A circle C is closed nowhere dense subset of the plane and determined by any three points, hence C ∩ G ⊆ span_{ℤ}(F) for some finite F ⊆ G. Also the graph Q of a polynomial of degree n is determined by any n+1 points so there will be a finite F ⊆ G with G ∩ Q ⊆ span_{ℤ}(F).

4. One can get a group G ⊆ ℝ^{2} which is continuum-dense and meets every line in a discrete set. To see this construct a sequence of perfect sets P_{n} such that every nonempty open set contains one of them and such that for every finite {p_{i} : i < n} ⊆ ∪_{n}P_{n} of distinct points the tuple (p_{i} : i < n) is ℙ^{n}-generic over M. Then G = span_{ℤ}(∪_{n}P_{n}) meets every line discretely.

Theorem 8 There exists a dense subgroup G ⊆ ℝ^{N} such that G ∩ V is discrete for every V ⊆ ℝ^{N} a vector space over ℝ of dimension N − 1.

Proof

In the model M we let ℙ be the nonempty open subsets of ℝ^{N}. Take (p_{i} : i < ω) be ∑_{n<ω} ℙ-generic over M and put G = span_{ℤ}({p_{n} : n < ω}).

Let ∑_{n<ω} ℤ be the sequences x ∈ ℤ^{ω} such that x(n) = 0 except for finitely many n. Let h : ∑_{n<ω} ℤ → ℝ^{N} be the homomorphism defined by

h(x) = ∑_{n<ω} x(n)p_{n}

and note that this is a finite sum, the coefficients are integers and the p_{n} are generic vectors in ℝ^{N}.

Clearly the generic vectors are linearly independent over the rationals so the map h is a group isomorphism h : ∑_{n<ω} ℤ → G.

Let C ⊆ ∑_{n<ω} ℤ be h^{−1}(G ∩ V). As in the proof of Theorem 4.2 Beros [1], let

C_{n} = {x ∈ C : ∀i < n x(i) = 0}.

If C_{n+1} is a proper subgroup of C_{n} choose k_{n} ∈ ℤ non-zero which divides every y(n) for y ∈ C_{n}. Otherwise, if C_{n+1} = C_{n} put k_{n} = 0. For each n choose x_{n} ∈ C_{n} with x_{n}(n) = k_{n}.

Claim. If k_{n} ≠ 0, then h(x_{n}) ∉ span_{ℝ}({h(x_{i}) : i > n}). Proof Suppose for contradiction that for some m

h(x_{n}) ∈ span_{ℝ}({h(x_{n+i}) : i = 1 … m}).

Let V_{0} = span_{ℝ}({h(x_{n+i}) : i = 1 … m}) and note that V_{0} ⊆ V and so dim(V_{0}) ≤ dim(V) < N. Hence V_{0} is closed and nowhere dense in ℝ^{N}.

Now ⟨h(x_{n+i}) : i = 1 … m⟩ ∈ M[⟨p_{i} : i > n⟩] and hence V_{0} is a closed subset of ℝ^{N} coded in M[⟨p_{i} : i > n⟩]. But

h(x_{n}) = k_{n}p_{n} + ∑_{i>n} x_{n}(i)p_{i}

and so as we saw in the last proof h(x_{n}) is ℙ-generic over M[⟨p_{i} : i > n⟩]. But this means that it avoids the closed nowhere dense set V_{0} coded in M[⟨p_{i} : i > n⟩]. Contradiction. QED

Claim. |{n : k_{n} ≠ 0}| ≤ dim(V). Proof Suppose not and let Q ⊆ {n : k_{n} ≠ 0} be finite with |Q| > dim(V). Then for each n ∈ Q we have that

h(x_{n}) ∉ span_{ℝ}({h(x_{m}) : m ∈ Q and m > n}).

It follows that {h(x_{n}) : n ∈ Q} is a linearly independent subset of V of size greater than dim(V), which is a contradiction.

QED

Hence Q =^{def} {n : k_{n} ≠ 0} is finite. This means that

C = span_{ℤ}({x_{n} : n ∈ Q})

by a process resembling Gaussian elimination. Knowing that Q is finite shows that the algorithm terminates.

It follows that V ∩ G = span_{ℤ}({h(x_{n}) : n ∈ Q}) and {h(x_{n}) : n ∈ Q} is a basis for span_{ℝ}(V ∩ G). Extend {h(x_{n}) : n ∈ Q} to a basis B for ℝ^{N}. Take a bijective linear transformation j : ℝ^{N} → ℝ^{N} determined by mapping the standard basis of ℝ^{N} to B. It follows that j(ℤ^{|Q|} × {0}) = V ∩ G and so V ∩ G is discrete in ℝ^{N}. QED

Remark 9 For ℝ^{ω} using this technique we get a dense generically generated group G ⊆ ℝ^{ω} such that V ∩G is discrete for every finite dimensional subspace V ⊆ ℝ^{ω}.

Although we are enamored with the generic sets argument of Theorem 8, it also possible to give an elementary inductive construction. We haven’t verified it in general but we did check the proof for the plane. As long as we are at it, we might as well construct a subgroup of ℝ^{2} which is maximal with respect to linear discreteness, although this will require a subgroup of cardinality the continuum.

Theorem 10 Let L be the family of all lines in the plane containing the origin. There exists a subgroup G ⊆ ℝ^{2} such that G∩L is an infinite discrete set for every L ∈ L.

Proof Define G ⊆ ℝ^{2} is linearly discrete iff G ∩ L is discrete for every L ∈ L. Define L_{G} = {L ∈ L : |G ∩ L| ≥ 2}.

Lemma 11 For any linearly discrete subgroup G ⊆ ℝ^{2} with |G| < |ℝ| and any L_{0} ∈ L \ L_{G} there exists p ∈ L_{0} \ {0} such that G + ℤp (the group generated by G and p) is linearly discrete and (G+ℤp)∩L = G∩L for every L ∈ L_{G}.

Proof Suppose np + q ∈ L where n ∈ ℤ is nonzero, L ∈ L_{G}, and q ∈ G. Then p + \frac{q}{n} ∈ L since L contains the origin. But the lines L_{0} and L − \frac{q}{n} meet in at most one point. It follows that at most |G| many points of L_{0} are ruled out so we may choose p ∈ L_{0} \ {0} such (G + ℤp) ∩ L = G ∩ L for every L ∈ L_{G}.

This implies that G + ℤp is linearly discrete by the following argument. We only need to worry about new lines L ∈ L which are not in L_{G}. Suppose np+q ∈ L where L ∈ L\L_{G}, n ∈ ℤ nonzero, and q ∈ G. Choose such a point on L with minimal n > 0. We claim that L ∩ (G + pℤ) = {knp + kq; k ∈ ℤ}.

Then for any k ∈ ℤ we have that knp+kq ∈ L. Any point on L∩(G+pℤ) has the form Let mp + q′ be an arbitrary point of pℤ + G. If it is in L it must be that n divides m otherwise 0 < m − kn < n for some k ∈ ℤ and

(mp + q′) − (knp + kq) = (m − kn)p + (q′ − kq) ∈ L

shows that n was not minimal. Hence for some k ∈ ℤ we have that kn = m. It must be that kq = q′ since otherwise q′ − kq ∈ L is a nontrivial element of G in L, contradicting L ∉ L_{G}. QED

Using the Lemma it is an easy inductive construction of an and increasing sequence of subgroups G_{α} ⊆ ℝ^{2} for α < |ℝ| such that |G_{α}| = |α+ω| is linearly discrete and for each line L ∈ L there is an α with |G_{α} ∩ L| ≥ 2. QED

Question 12 For a line L in the plane not containing the origin, if G meets L at all, then taking u ∈ G ∩ L and considering L − u ∈ L shows us that G meets L in an infinite discrete set. I don’t know if we can construct such a G which meets every line. I don’t even know how to meet all of the horizontal lines.

Jan Pachl points out that every locally compact compactly generated Abelian group is a product of a compact Abelian group and a finite number of copies of ℝ and ℤ (see Hewitt and Ross [2] 9.8). Also, every locally compact Abelian group is a product of ℝ^{k} and a locally compact subgroup that has a compact open subgroup (see Hofmann and Morris [3] 7.57). Every K_{σ} topological group is a closed subgroup of a compactly generated group (see Pestov [4]).

References

[1] Universal subgroups of Polish groups, Konstantinos A. Beros

http://www.math.wisc.edu/~kberos/research.html
http://arxiv.org/abs/1209.3497

[2] Hewitt, Edwin; Ross, Kenneth A.;Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations. Die Grundlehren der mathematischen Wissenschaften, Bd. 115 Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Gttingen-Heidelberg, 1963 viii+519 pp.

[3] Hofmann, Karl H.; Morris, Sidney A.; The structure of compact groups. A primer for the student, a handbook for the expert. de Gruyter Studies in Mathematics, 25. Walter de Gruyter & Co., Berlin, 1998. xviii+835 pp. ISBN: 3-11-015268-1

[4] Pestov, V. G.; On compactly generated topological groups. (Russian) Mat. Zametki 40 (1986), no. 5, 671-676, 699. English translation: Math. Notes 40 (1986), 880–882.

Arnold W. Miller miller@math.wisc.edu http://www.math.wisc.edu/∼miller University of Wisconsin-Madison Department of Mathematics, Van Vleck Hall 480 Lincoln Drive Madison, Wisconsin 53706-1388