Addendum to "Tilings problems on Baumslag-Solitar groups"
aa r X i v : . [ m a t h . G R ] J a n Addendum to ”
Tilings problems onBaumslag-Solitar groups ” Nathalie Aubrun and Jarkko Kari CNRS, Universit´e Paris-Saclay, LISN, 91400 Orsay, France Department of Mathematics, University of Turku, FIN-20014, Turku, Finland
Abstract
In our article [AK13] we state the the Domino problem is undecidablefor all Baumslag-Solitar groups BS ( m, n ), and claim that the proof isa direct adaptation of the construction of a weakly aperiodic subshiftof finite type for BS ( m, n ) given in the paper. In this addendum, weclarify 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 Dominoproblem is undecidable on all Baumslag-Solitar groups BS ( m, n ). It turns outthat it is not as immediate as we write it, and we believe that this result deservesa full explanation.The proof is based on the proof of the undecidability of the Domino problemon the discrete hyperbolic plane given by the second author in [Kar07].Thislatter is an adaptation of a former construction of a strongly aperiodic SFTon Z [Kar96]. This proof proceeds by reduction to the immortality problemfor rational piecewise affine maps. We first recall the key ingredients of thisproof.A mapping f : U → U ⊂ R is a rational piecewise affine map if there existsa partition U = U ∪ U ∪ · · · ∪ U n where every U i is a unitary square withinteger coordinates, and such that f = f i on every U i , and f i : U i → R is anaffine function with rational parameters. A point −→ x ∈ U is immortal for f if forevery k ∈ Z , the iterated image f k ( −→ x ) belongs to U . The immortality problemfor rational piecewise affine maps is the decision problem that inputs such afunction f , and outputs Yes if f possesses an immortal point, and No otherwise.This problem reduces to the immortality problem for Turing machines, whichis known to be undecidable [Hoo66]. Theorem 1 ([Kar07]) . The immortality problem for rational piecewise affinemaps is undecidable. H with pentagonal Wang tiles is undecidable, by a reductionto the immortality problem for rational piecewise affine maps. The proof isbased on the construction, for every piecewise affine map f : U → U withrational parameters, of a finite tileset that computes the function f , meaningthat a tiling by this tileset encodes the orbit of a point −→ x ∈ U under the actionof f . Theorem 2 ([Kar07]) . The Domino problem is undecidable on the discretehyperbolic plane H . The most difficult part in the construction is to ensure finiteness of the tileset.This issue may be bypassed by combining two main ingredients: representingreal numbers by Beatty sequences, and taking advantage of the rationality offunction f . Instead of Beatty sequences of a point −→ x ∈ R we use its balancedrepresentation, to be defined on page 6, and that takes accounts of the mergesbetween sheets in BS ( m, n ). Adaptation to Baumslag-Solitar groups BS ( m, n ) Following [Kar07] we prove that the Domino problem is undecidable for allBaumslag-Solitar BS ( m, n ) for all integers m, n ∈ N ∗ BS ( m, n ) = h a, t | t − a m t = a n i . Since BS ( − m, − n ) is isomorphic to BS ( m, n ), it is enough to considergroups with m >
0. For simplicity, we also assume that n >
0. The case n < BS ( m, n ) with generating set { a, t, a − , t − } is made of several sheets that merge m by m from top to give n other sheets tothe 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 hyperbolicplane H . t a ata a a Figure 1: A portion of the right Cayley graph of the group BS (2 ,
3) with gen-erating set { a, t, a − , t − } . Three sheets from the top merge and separate intotwo sheets to the bottom. 2he group BS ( m, n ) is embedded into R through a function Φ m,n : BS ( m, n ) → R , that is first defined recursively on words w on the alphabet A = { a, t, a − , t − } .If x is a letter from A , denote | w | x the number of occurrences of the letter x inthe word w . We also call contribution of x to w the integer k w k x = | w | x −| w | x − .With these notations we first define a function β : A ∗ → Z by β ( w ) := − k w k t ,and a function α m,n : A ∗ → R , simply denoted α in the sequel, which is definedby induction on the length of words ( ε denotes the empty word) by: α ( ε ) = 0 α ( w.t ) = α ( w.t − ) = α ( w ) α ( w.a ) = α ( w ) + (cid:16) mn (cid:17) − β ( w ) α ( w.a − ) = α ( w ) − (cid:16) mn (cid:17) − β ( w ) By induction on the length of words w we get the formula: Proposition 3.
For every words u, v ∈ A ∗ one has α ( u.v ) = α ( u ) + (cid:16) mn (cid:17) − β ( u ) α ( v ) . In the sequel we will use in particular the following equalities: α ( ga ) = α ( g ) + (cid:16) mn (cid:17) − β ( g ) β ( gt ) = β ( g ) − m,n : BS ( m, n ) → R isΦ m,n ( g ) = ( α ( w ) , β ( w )) , where w is a word that represents the group element g . One can check thatΦ m,n is well-defined, i.e. the value for Φ m,n ( g ) does not depend on the word w chosen, thanks to Proposition 3. Proposition 4.
The function Φ m,n is well-defined on BS ( m, n ) . Note that for m = 1 we find the same function as the isomorphism Φ definedonly for amenable Baumslag-Solitar groups in [AS20]. Remark 5. If | m | 6 = 1 and | n | 6 = 1 then the function Φ m,n is not injective.In [AK13] we give an example of injectivity default for m = 3 and n = 2 : thegroup element ω = bab − a ba − b − a − is sent to the origin by Φ , , but hasinfinite order. In [EM20] the authors exhibit the word ω = bab − aba − b − a − which satisfies that Φ m,n ( ω ) = (0 , for all m, n such that | m | 6 = 1 and | n | 6 = 1 . → x . . . −→ x n −→ ℓ −→ r −→ y −→ y . . . −→ y m Figure 2: A Wang tile for BS ( m, n ).Fix two integers m, n ∈ N ∗ . A tile on BS ( m, n ) computes a function f i : U i ⊂ R → R if, the following holds (colors on the edges of the tile are namedafter Figure 2): −→ y + . . . −→ y m m + −→ r = f i (cid:18) −→ x + . . . −→ x n n (cid:19) + −→ ℓ . For a single sheet of BS ( m, n ), one can easily adapt what is done for thediscrete hyperbolic plane H [Kar07]: select all tiles satisfying the relation withcolors on the edges belonging to a well chosen finite set. For the whole groupdifficulties arise where different sheets merge: clearly the direct adaptation ofthe H case is not enough, and the tileset should be enriched to take into accountthe specific structure of BS ( m, n ). Our solution uses function Φ m,n , from whichwe define a function λ : BS ( m, n ) → R by λ ( g ) := 1 m (cid:16) nm (cid:17) − β ( g ) α ( g ) , for every g ∈ BS ( m, n ), and one can check that the following holds λ ( ga ) = λ ( g ) + 1 mλ ( gt ) = nm λ ( g ) . Thanks to the function λ and to the properties it satisfies, we detail thecontent of every tile that computes a piecewise affine function f i : U i ⊂ R → R such that f i ( −→ x ) = M −→ x + −→ b : −→ x k ( g, −→ x ) := ⌊ ( nλ ( g ) + k ) −→ x ⌋ − ⌊ ( nλ ( g ) + ( k − −→ x ⌋ for k = 1 . . . m −→ y k ( g, −→ x ) := ⌊ ( mλ ( g ) + k ) f i ( −→ x ) ⌋ − ⌊ ( mλ ( g ) + ( k − f i ( −→ x ) ⌋ for k = 1 . . . n −→ ℓ ( g, −→ x ) := 1 n f i ( ⌊ nλ ( g ) −→ x ⌋ ) − m ⌊ mλ ( g ) f i ( −→ x ) ⌋ + ⌊ λ ( g ) − ⌋−→ b −→ r ( g, −→ x ) := 1 n f i ( ⌊ ( nλ ( g ) + n ) −→ x ⌋ ) − m ⌊ ( mλ ( g ) + m ) f i ( −→ x ) ⌋ + ⌊ λ ( g ) + 12 ⌋−→ b We check that the tile on Figure 3 does compute the function f i , in other4 ( g, −→ x ) . . . −→ x n ( g, −→ x ) −→ ℓ ( g, −→ x ) −→ r ( g, −→ x ) −→ y ( g, −→ x ) −→ y ( g, −→ x ) . . . −→ y m ( g, −→ x ) • g Figure 3: Tile to encode a piecewise affine map f i : U i ⊂ R → R on the group BS ( m, n ).words that the quantity S detailed below sums to null vector −→ S := −→ y + · · · + −→ y m m + −→ r − f i (cid:18) −→ x + · · · + −→ x n n (cid:19) − −→ ℓ By replacing every term −→ y k and −→ x k by its expression given above, the twosums −→ y + · · · + −→ y m and −→ x + · · · + −→ x n telescope and S simplifies in S = 1 m ⌊ ( mλ ( g ) + m ) f i ( −→ x ) ⌋ − m ⌊ mλ ( g ) f i ( −→ x ) ⌋ + 1 n f i ( ⌊ ( nλ ( g ) + n ) −→ x ⌋ ) − m ⌊ ( mλ ( g ) + m ) f i ( −→ x ) ⌋ + ⌊ λ ( g ) + 12 ⌋−→ b − f i (cid:18) n ⌊ ( nλ ( g ) + n ) −→ x ⌋ − n ⌊ nλ ( g ) −→ x ⌋ (cid:19) − n f i ( ⌊ nλ ( g ) −→ x ⌋ ) + 1 m ⌊ mλ ( g ) f i ( −→ x ) ⌋ − ⌊ λ ( g ) − ⌋−→ b which then reduces to S = 1 n f ( ⌊ ( nλ ( g ) + n ) −→ x ⌋ ) + ⌊ λ ( g ) + 12 ⌋−→ b − f (cid:18) n ⌊ ( nλ ( g ) + n ) −→ x ⌋ − n ⌊ nλ ( g ) −→ x ⌋ (cid:19) − n f ( ⌊ nλ ( g ) −→ x ⌋ ) − ⌊ λ ( g ) − ⌋−→ b . We now use the fact that f i ( c −→ y − c −→ z ) = cf i ( −→ y ) − cf i ( −→ z ) + −→ b to obtain: S = − n f i ( ⌊ nλ ( g ) −→ x ⌋ ) + ⌊ λ ( g ) + 12 ⌋−→ b − n f i ( ⌊ ( nλ ( g ) + n ) −→ x ⌋ ) + 1 n f i ( ⌊ nλ ( g ) −→ x ⌋ ) − −→ b + 1 n f i ( ⌊ ( nλ ( g ) + n ) −→ x ⌋ ) − ⌊ λ ( g ) − ⌋−→ b S = ⌊ λ ( g ) + 12 ⌋−→ b − −→ b − ⌊ λ ( g ) − ⌋−→ b and since ⌊ z + ⌋ − ⌊ z − + ⌋ = 1 for every real number z , we finally concludethat S = 0. Proposition 6.
For every g ∈ BS ( m, n ) and every −→ x ∈ U i , the tile describedon Figure 3 computes the piecewise affine map f i : U i ⊂ R → R . Every tile thus individually computes the image by f i of the average of theelements 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 rightedges of the tile.As explained in [AK13], for all −→ x ∈ R and z ∈ R , if one defines for every k ∈ Z −→ B k ( −→ x , z ) := ⌊ ( z + k ) −→ x ⌋ − ⌊ ( z + ( k − −→ x ⌋ , then the bi-infinite sequence (cid:16) −→ B k ( −→ x , z ) (cid:17) k ∈ Z is a balanced representation of −→ x .In particular, it is a representation of −→ x = ( x , x ), meaning that • every −→ B k ( −→ x , z ) has integer coordinates in {⌊ x ⌋ ; ⌊ x ⌋ + 1 }×{⌊ x ⌋ ; ⌊ x ⌋ + 1 } ; • the following average converges towards −→ x lim k →∞ k + 1 k X j = − k −→ B j ( −→ x , z ) = −→ x . Proposition 7.
For every g ∈ BS ( m, n ) and every −→ x ∈ U i , if we put thetile from Figure 3 in position g for every g ∈ (cid:8) g · a k | k ∈ Z (cid:9) , then one canread the balanced representation −→ B k ( −→ x , λ ( g )) of −→ x on the bottom edges and thebalanced representation −→ B k (cid:0) f i ( −→ x ) , λ ( gt − ) (cid:1) of f i ( −→ x ) on the top edges. Proposition 7 expresses the fact that moving from a single tile that computes f i with errors to an infinite row of tiles makes the calculation of f i exact.We now check that among all possible tiles that compute f i , we can restrict toa finite tileset. For every g ∈ BS ( m, n ), the sequence −→ B k ( −→ x , λ ( g )) is a balancedrepresentation of −→ x , so that there exist only finitely many possible values for −→ x k , and the same argument prevails for −→ y k . It remains to check that the −→ ℓ and −→ r can be chosen among a finite set. Using the fact that λ ( ga m ) = λ ( g ) + 1,we remark that −→ ℓ ( ga m , −→ x ) = −→ r ( g, −→ x ) ; hence it is enough to ensure a finitenumber of choices for the −→ ℓ only. Remind that6 → ℓ ( g, −→ x ) := 1 n f i ( ⌊ nλ ( g ) −→ x ⌋ ) − m ⌊ mλ ( g ) f i ( −→ x ) ⌋ + ⌊ λ ( g ) − ⌋−→ b . We first check that −→ ℓ ( g, −→ x ) is bounded, as a consequence of −→ z − ≤ ⌊−→ z ⌋ < −→ z (inequalities shall apply coordinate by coordinate). Indeed: λ ( g ) (cid:16) f i ( −→ x ) − −→ b (cid:17) − n M −→ − n −→ b − λ ( g ) f i ( −→ x ) + λ ( g ) −→ b − − −→ b + −→ b < −→ ℓ ( g, −→ x ) − n M −→ − n + 22 n −→ b < −→ ℓ ( g, −→ x )and −→ ℓ ( g, −→ x ) < λ ( g ) (cid:16) f i ( −→ x ) − −→ b (cid:17) + 1 n −→ b − λ ( g ) f i ( −→ x ) + λ ( g ) −→ b − − m −→ λ ( g ) −→ b − −→ b −→ ℓ ( g, −→ x ) < − m −→ − n − n −→ b . Since both vector −→ b and matrix M have rational coefficients, one can putat the same denominator q the two inequalities above, so that there exist twovectors −→ p , −→ p ∈ Z such that −→ p q ≤ −→ ℓ ( g, −→ x ) ≤ −→ p q , where −→ p is chosen maximal and −→ p minimal. Better than that, the value for −→ ℓ ( g, −→ x ) should belong to the finite set ( −→ p q , −→ p + (0 , q , −→ p + (1 , q , −→ p + −→ q , . . . , −→ p q ) ⊂ Q for every g ∈ BS ( m, n ) and every −→ x ∈ U . Indeed, a careful observation ofrational numbers that appear in the expression of −→ ℓ ( g, −→ x ), shows that −→ ℓ ( g, −→ x )can be written as −→ pq . The fact that −→ p ≤ −→ p ≤ −→ p directly follows from thedefinition of −→ p and −→ p . The tileset τ f i corresponding to the function f i is thusfinite. Proposition 8.
There exists a finite number of tiles on BS ( m, n ) with colorsas on Figure 3 that computes f i ( −→ x ) for every −→ x ∈ U i . Thanks to the properties of the function λ stated above, one has y (cid:0) g · a k , −→ x (cid:1) = y k ( g, −→ x ) for k ∈ [1; m − y ( gt, −→ x ) = x ( g, f i ( −→ x )) , −→ x ∈ U i , there exists a tiling of the coset { a k | k ∈ Z } such that the balanced representations of −→ x and f i ( −→ x ) appear respectivelyon bottom and top edges. We then put together all tilesets corresponding toevery function f i , by adding to these tiles the number i of the function f i theyencode. With the additional local rule that two tiles in positions g and ga shouldshare the same number i , we finally get the desired result. Theorem 9.
The Domino problem is undecidable on Baumslag-Solitar groups BS ( m, n ) for every integers m, n ∈ Z . Acknowledgments
This work was partially supported by the ANR project CoCoGro (ANR-16-CE40-0005).
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