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Addendum to ”Tilings problems on Baumslag-Solitar groups”
Nathalie Aubrun, Jarkko Kari
To cite this version:
Nathalie Aubrun, Jarkko Kari. Addendum to ”Tilings problems on Baumslag-Solitar groups”. 2021.
�hal-03126862�
arXiv:2101.12470v1 [math.GR] 29 Jan 2021
Addendum to ” Tilings problems on Baumslag-Solitar groups ”
Nathalie Aubrun
1and Jarkko Kari
21CNRS, Universit´e Paris-Saclay, LISN, 91400 Orsay, France
2Department of Mathematics, University of Turku, FIN-20014, Turku, Finland
Abstract
In our article [AK13] we state the the Domino problem is undecidable for all Baumslag-Solitar groups BS(m, n), and claim that the proof is a direct adaptation of the construction of a weakly aperiodic subshift of finite type for BS(m, n) given in the paper. In this addendum, we clarify this point and give a detailed proof of the undecidability result.
We assume the reader is already familiar with the article [AK13].
Introduction
In [AK13] we state as a direct corollary of the main construction that the Domino problem is undecidable on all Baumslag-Solitar groupsBS(m, n). It turns out that it is not as immediate as we write it, and we believe that this result deserves a full explanation.
The proof is based on the proof of the undecidability of the Domino problem on the discrete hyperbolic plane given by the second author in [Kar07].This latter is an adaptation of a former construction of a strongly aperiodic SFT onZ2 [Kar96]. This proof proceeds by reduction to the immortality problem for rational piecewise affine maps. We first recall the key ingredients of this proof.
A mappingf :U →U ⊂R2is a rational piecewise affine map if there exists a partition U = U1∪U2∪ · · · ∪Un where every Ui is a unitary square with integer coordinates, and such thatf =fi on everyUi, andfi :Ui →R2 is an affine function with rational parameters. A point−→x ∈U is immortal forf if for everyk∈Z, the iterated imagefk(−→x) belongs toU. The immortality problem for rational piecewise affine maps is the decision problem that inputs such a functionf, and outputsYesiff possesses an immortal point, andNootherwise.
This problem reduces to the immortality problem for Turing machines, which is known to be undecidable [Hoo66].
Theorem 1 ([Kar07]). The immortality problem for rational piecewise affine maps is undecidable.
The second author proves in [Kar07] that the problem of tiling the discrete hyperbolic planeH2 with pentagonal Wang tiles is undecidable, by a reduction to the immortality problem for rational piecewise affine maps. The proof is based on the construction, for every piecewise affine map f : U → U with rational parameters, of a finite tileset that computes the functionf, meaning that a tiling by this tileset encodes the orbit of a point−→x ∈U under the action off.
Theorem 2 ([Kar07]). The Domino problem is undecidable on the discrete hyperbolic planeH2.
The most difficult part in the construction is to ensure finiteness of the tileset.
This issue may be bypassed by combining two main ingredients: representing real numbers by Beatty sequences, and taking advantage of the rationality of functionf. Instead of Beatty sequences of a point−→x ∈R2 we use its balanced representation, to be defined on page 6, and that takes accounts of the merges between sheets inBS(m, n).
Adaptation to Baumslag-Solitar groups BS ( m, n )
Following [Kar07] we prove that the Domino problem is undecidable for all Baumslag-SolitarBS(m, n) for all integersm, n∈N∗
BS(m, n) =ha, t|t−1amt=ani.
Since BS(−m,−n) is isomorphic to BS(m, n), it is enough to consider groups withm >0. For simplicity, we also assume thatn >0. The casen <0 is analogous. The Cayley graph ofBS(m, n) with generating set{a, t, a−1, t−1} is made of several sheets that mergembymfrom top to givenother sheets to the bottom, so that the global structure these sheets are arranged looks like an (m+n)-regular tree. Each of these sheets is quasi-isometric to the hyperbolic planeH2.
t
a a
t
a a a
Figure 1: A portion of the right Cayley graph of the groupBS(2,3) with gen- erating set{a, t, a−1, t−1}. Three sheets from the top merge and separate into two sheets to the bottom.
The groupBS(m, n) is embedded intoR2through a function Φm,n:BS(m, n)→ R2, that is first defined recursively on wordswon the alphabetA={a, t, a−1, t−1}.
Ifxis a letter fromA, denote|w|xthe number of occurrences of the letterxin the wordw. We also callcontributionofxtowthe integerkwkx=|w|x−|w|x−1. With these notations we first define a functionβ:A∗ →Zbyβ(w) :=− kwkt, and a functionαm,n:A∗→R, simply denotedαin the sequel, which is defined by induction on the length of words (εdenotes the empty word) by:
α(ε) = 0
α(w.t) =α(w.t−1) =α(w) α(w.a) =α(w) +m
n −β(w)
α(w.a−1) =α(w)−m n
−β(w)
By induction on the length of wordswwe get the formula:
Proposition 3. For every words u, v∈A∗ one has
α(u.v) =α(u) +m n
−β(u) α(v).
In the sequel we will use in particular the following equalities:
α(ga) =α(g) +m n
−β(g)
β(gt) =β(g)−1 Finally the function Φm,n:BS(m, n)→R2 is
Φm,n(g) = (α(w), β(w)),
where w is a word that represents the group element g. One can check that Φm,n is well-defined, i.e. the value for Φm,n(g) does not depend on the wordw chosen, thanks to Proposition 3.
Proposition 4. The function Φm,nis well-defined on BS(m, n).
Note that form= 1 we find the same function as the isomorphism Φ defined only for amenable Baumslag-Solitar groups in [AS20].
Remark 5. If |m| 6= 1 and |n| 6= 1 then the function Φm,n is not injective.
In [AK13] we give an example of injectivity default form = 3and n= 2: the group element ω = bab−1a2ba−1b−1a−2 is sent to the origin byΦ3,2, but has infinite order. In [EM20] the authors exhibit the wordω =bab−1aba−1b−1a−1 which satisfies thatΦm,n(ω) = (0,0) for allm, nsuch that|m| 6= 1and|n| 6= 1.
−
→x1 . . . −→ xn
−
→ℓ −→
r
−
→y1 −→
y2 . . . −→ ym
Figure 2: A Wang tile forBS(m, n).
Fix two integers m, n ∈N∗. A tile on BS(m, n) computes a function fi : Ui⊂R2→R2 if, the following holds (colors on the edges of the tile are named after Figure 2):
−
→y1+. . .−→ym
m +−→r =fi
−→x1+. . .−→xn
n
+−→ ℓ .
For a single sheet of BS(m, n), one can easily adapt what is done for the discrete hyperbolic planeH2[Kar07]: select all tiles satisfying the relation with colors on the edges belonging to a well chosen finite set. For the whole group difficulties arise where different sheets merge: clearly the direct adaptation of theH2case is not enough, and the tileset should be enriched to take into account the specific structure ofBS(m, n). Our solution uses function Φm,n, from which we define a functionλ:BS(m, n)→Rby
λ(g) := 1 m
n m
−β(g)
α(g),
for everyg∈BS(m, n), and one can check that the following holds λ(ga) =λ(g) + 1
m λ(gt) = n
mλ(g).
Thanks to the function λ and to the properties it satisfies, we detail the content of every tile that computes a piecewise affine functionfi:Ui⊂R2→R2 such thatfi(−→x) =M−→x +−→
b:
−
→xk(g,−→x) :=⌊(nλ(g) +k)−→x⌋ − ⌊(nλ(g) + (k−1))−→x⌋ fork= 1. . . m
−
→yk(g,−→x) :=⌊(mλ(g) +k)fi(−→x)⌋ − ⌊(mλ(g) + (k−1))fi(−→x)⌋ fork= 1. . . n
−
→ℓ(g,−→x) := 1
nfi(⌊nλ(g)−→x⌋)− 1
m⌊mλ(g)fi(−→x)⌋+⌊λ(g)−1 2⌋−→
b
−
→r(g,−→x) := 1
nfi(⌊(nλ(g) +n)−→x⌋)− 1
m⌊(mλ(g) +m)fi(−→x)⌋+⌊λ(g) +1 2⌋−→
b We check that the tile on Figure 3 does compute the function fi, in other
x1(g,−→
x) . . . −→
xn(g,−→ x)
−
→ ℓ (g,−→
x) −→
r (g,−→ x)
−
→y1(g,−→ x) −→
y2(g,−→
x) . . . −→
ym(g,−→ x)
g •
Figure 3: Tile to encode a piecewise affine mapfi:Ui⊂R2→R2on the group BS(m, n).
words that the quantityS detailed below sums to null vector−→ 0 : S:=
−
→y1+· · ·+−→ym
m +−→r −fi
−→x1+· · ·+−→xn
n
−−→ ℓ
By replacing every term−→yk and−→xk by its expression given above, the two sums−→y1+· · ·+−→ymand−→x1+· · ·+−→xn telescope andS simplifies in
S= 1
m⌊(mλ(g) +m)fi(−→x)⌋ − 1
m⌊mλ(g)fi(−→x)⌋+1
nfi(⌊(nλ(g) +n)−→x⌋)− 1
m⌊(mλ(g) +m)fi(−→x)⌋
+⌊λ(g) +1 2⌋−→
b −fi
1
n⌊(nλ(g) +n)−→x⌋ −1
n⌊nλ(g)−→x⌋
−1
nfi(⌊nλ(g)−→x⌋) + 1
m⌊mλ(g)fi(−→x)⌋ − ⌊λ(g)−1 2⌋−→
b which then reduces to
S = 1
nf(⌊(nλ(g) +n)−→x⌋) +⌊λ(g) +1 2⌋−→
b
−f 1
n⌊(nλ(g) +n)−→x⌋ −1
n⌊nλ(g)−→x⌋
−1
nf(⌊nλ(g)−→x⌋)− ⌊λ(g)−1 2⌋−→
b .
We now use the fact thatfi(c−→y −c−→z) =cfi(−→y)−cfi(−→z) +−→
b to obtain:
S=−1
nfi(⌊nλ(g)−→x⌋) +⌊λ(g) +1 2⌋−→
b
−1
nfi(⌊(nλ(g) +n)−→x⌋) +1
nfi(⌊nλ(g)−→x⌋)−−→ b + 1
nfi(⌊(nλ(g) +n)−→x⌋)− ⌊λ(g)−1 2⌋−→
b
afterwards only remains:
S=⌊λ(g) +1 2⌋−→
b −−→
b − ⌊λ(g)−1 2⌋−→
b
and since⌊z+12⌋ − ⌊z−+12⌋= 1 for every real numberz, we finally conclude thatS= 0.
Proposition 6. For every g∈BS(m, n)and every −→x ∈Ui, the tile described on Figure 3 computes the piecewise affine mapfi:Ui⊂R2→R2.
Every tile thus individually computes the image byfi of the average of the elements on the bottoms edges, and redistributes this image on the top edges.
This is performed up to calculation errors, that are stored in the left and right edges of the tile.
As explained in [AK13], for all−→x ∈R2 and z∈R, if one defines for every k∈Z
−
→Bk(−→x , z) :=⌊(z+k)−→x⌋ − ⌊(z+ (k−1))−→x⌋, then the bi-infinite sequence−→
Bk(−→x , z)
k∈Zis abalanced representation of−→x. In particular, it is a representation of−→x = (x1, x2), meaning that
• every−→
Bk(−→x , z) has integer coordinates in{⌊x1⌋;⌊x1⌋+ 1}×{⌊x2⌋;⌊x2⌋+ 1}
;
• the following average converges towards−→x
klim→∞
1 2k+ 1
k
X
j=−k
−
→Bj(−→x , z) =−→x .
Proposition 7. For every g0 ∈ BS(m, n) and every −→x ∈ Ui, if we put the tile from Figure 3 in position g for every g ∈
g0·ak|k∈Z , then one can read the balanced representation−→
Bk(−→x , λ(g))of−→x on the bottom edges and the balanced representation−→
Bk fi(−→x), λ(gt−1)
offi(−→x)on the top edges.
Proposition 7 expresses the fact that moving from a single tile that computes fi with errors to an infinite row of tiles makes the calculation offi exact.
We now check that among all possible tiles that computefi, we can restrict to a finite tileset. For everyg∈BS(m, n), the sequence−→
Bk(−→x , λ(g)) is a balanced representation of−→x, so that there exist only finitely many possible values for
−
→xk, and the same argument prevails for −→yk. It remains to check that the−→ ℓ and−→r can be chosen among a finite set. Using the fact thatλ(gam) =λ(g) + 1, we remark that−→
ℓ(gam,−→x) =−→r(g,−→x) ; hence it is enough to ensure a finite number of choices for the−→
ℓ only. Remind that
−
→ℓ(g,−→x) := 1
nfi(⌊nλ(g)−→x⌋)− 1
m⌊mλ(g)fi(−→x)⌋+⌊λ(g)−1 2⌋−→
b .
We first check that−→
ℓ(g,−→x) is bounded, as a consequence of−→z −1≤ ⌊−→z⌋<
−
→z (inequalities shall apply coordinate by coordinate). Indeed:
λ(g)
fi(−→x)−−→ b
− 1 nM−→
1 − 1 n
−
→b −λ(g)fi(−→x) +λ(g)−→ b − −1
2
−
→b +−→ b <−→
ℓ(g,−→x)
−1 nM−→
1 −3n+ 2 2n
−
→b <−→ ℓ(g,−→x) and
−
→ℓ(g,−→x)< λ(g)
fi(−→x)−−→ b
+ 1 n
−
→b −λ(g)fi(−→x) +λ(g)−→ b − −1
m
−
→1 +λ(g)−→ b −1
2
−
→b
−
→ℓ(g,−→x)<−1 m
−
→1 −n−2 2n
−
→b .
Since both vector−→
b and matrix M have rational coefficients, one can put at the same denominatorq the two inequalities above, so that there exist two vectors−→p1,−→p2∈Z2 such that
−
→p1
q ≤−→
ℓ(g,−→x)≤
−
→p2
q ,
where −→p1 is chosen maximal and −→p2 minimal. Better than that, the value for
−
→ℓ(g,−→x) should belong to the finite set (−→p1
q ,
−
→p1+ (0,1)
q ,
−
→p1+ (1,0)
q ,
−
→p1+−→ 1 q , . . . ,
−
→p2
q )
⊂Q
for every g ∈ BS(m, n) and every −→x ∈ U. Indeed, a careful observation of rational numbers that appear in the expression of−→
ℓ(g,−→x), shows that−→ ℓ(g,−→x) can be written as →−pq . The fact that −→p1 ≤ −→p ≤ −→p2 directly follows from the definition of−→p1 and−→p2. The tilesetτfi corresponding to the functionfi is thus finite.
Proposition 8. There exists a finite number of tiles on BS(m, n) with colors as on Figure 3 that computesfi(−→x)for every−→x ∈Ui.
Thanks to the properties of the functionλstated above, one has y1 g·ak,−→x
=y1+k(g,−→x) fork∈[1;m−1]
y1(gt,−→x) =x1(g, fi(−→x)),
which ensures that for a given−→x ∈Ui, there exists a tiling of the coset{ak |k∈ Z}such that the balanced representations of−→x andfi(−→x) appear respectively on bottom and top edges. We then put together all tilesets corresponding to every functionfi, by adding to these tiles the numberiof the functionfi they encode. With the additional local rule that two tiles in positionsgandgashould share the same numberi, we finally get the desired result.
Theorem 9. The Domino problem is undecidable on Baumslag-Solitar groups BS(m, n)for every integersm, n∈Z.
Acknowledgments
This work was partially supported by the ANR project CoCoGro (ANR-16- CE40-0005).
References
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[AS20] Nathalie Aubrun and Michael Schraudner. Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on baumslag- solitar groupsBS(1, n). https://arxiv.org/abs/2012.11037, 2020.
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