On Limit Aperiodic G-Sets

ON LIMIT APERIODIC G-SETS THANOS GENTIMIS Abstract. W e pr o v e that the prop erty to b e limi t aperio dic is preserved by the sta ndard c onstruction with groups like extension, HNN extension and free product. W e also construct a non-li mit ap erio dic G-space. 1. Introduction If a discr ete gro up G acts by isometr ies freely and coco mpactly on a metric space X one ca n study p e rio dic and ap erio dic tilings of X . A tiling of X can b e defined first as a tiling with one tile, the V orono i cell (see [1 ]). Using a finite set of co lors one ca n consider tilings of X by color. The n using the ” notching” o ne can switch from a tiling by co lor to a g eometric tiling. The standar d exa mple here is G = Z 2 and X = R 2 . Note tha t the gr oup G in the a bove tilings is in bijection with the tiles. Thus, co ns truction o f a geometr ic tiling on X can b e reduced to a co loring o f the gro up G . In this pa pe r w e study the co lo rings of discre te groups G that lead to limit ap erio dic tilings. Let b ∈ G , a co loring φ of a group G is b -p erio dic if it is inv ariant under trans- lation b y b , i.e., for every element g ∈ G the element s g and bg have the sa me color. A co loring φ is ap erio dic if it is not b -p erio dic for a ny b ∈ G \ { e } . T his can be rephra sed as ”The s tabilizer of φ in the space of all color ings o f G is trivial”. F or infinite groups there is a strong notion of per io dicity: A coloring φ is strongly G -pe r io dic if | O r b G ( φ ) | < ∞ . The corresp onding neg ation called ’weakly ap er io dic’ means that the orbit O rb G ( φ ) = G/S tab G ( φ ) of φ is finite. A color ing φ is called (w eakly) limit ap erio dic if all colo rings in the clos ur e of the or bit Or b G ( φ ) taken in an a ppropriate s pa ce o f all coloring s are (weakly) ap erio dic. In this pap er we co nsider the question r aised in [1]: Whi ch gr oups admit limit ap erio dic c olorings by finitely many c olors? This is no t obvious que s tion even for G = Z . In [1] it was answered positively for torsio n free hyperb olic groups, Coxeter groups, a nd gr oups co mensurable to them. This ques tio n can b e sta ted in terms of T op ologic al Dynamica l Systems theory: L et G b e a gr oup and F b e a finite set. Do es the natur al action of G on the Cantor set F G admit a G -invariant c omp act subset X ⊂ F G such that the action of G on X is fr e e? The dyna mical system refor m ulation of a co rresp onding questio n ab out limit weak ap erio dic coloring s asks ab out a G -inv ariant compact subset X ⊂ F G such that the orbits O rb G ( x ) are infinite for all x ∈ X . This was answered affirmatively b y V. Usp e ns kii [1]. Moreov er, E. Glasner proved that there is a minimal set X ⊂ F G and x ∈ X with the trivia l stabilizer S tab G ( x ) = e . Despite on this progr ess the main ques tio n is s till ope n. In this pap er we give a group theoretic a pproach. W e call a gr oup ’limit a p e rio dic’ (LA for short) if it admits a Date : October 30, 2018. Key wor ds and phr ases. li mit ap erio dic groups and G-sets. 1 2 THANOS GENTIMIS limit aper io dic coloring b y finitely man y colors. W e show that the simple group constructions like the pro duct, the extension, the HNN extension, and the free pro duct preser ve the LA pr op erty . T o prov e these facts we introduce the notion of LA G -s pace and prov e the action theor em. In the end of the pap er we show that the main que s tion has a negative answer for a sp ecific G -set (the natura l num b er s) where G is a f.g. subgro up of Aut ( Z ). 2. Limit A periodic Groups Definition 2.1. Let G b e a f.g. gr o up. Also, let F be a finite set of elemen ts whic h we ca n think of as colo r s. A map φ from G to F is calle d a c oloring o f G . Definition 2.2. Let G , F b e as a b ove. W e denote by F G the s et of all colorings from G to F . If w e co nsider F with the discrete topolo gy , F G with the product top ology b eco mes a top olog ical space homeomor phic to the Cantor set. Definition 2.3. Let G , F a s ab ov e. Then G acts on F G with the left action δ : G × F G → F G defined by the fo r mula ( g ∗ f )( a ) = f ( g − 1 · a ) for every g , a ∈ G and f ∈ F G . Since F G is metrizable, a function φ b elongs to the closur e o f the orbit of f , φ ∈ Or b G ( f ), if and o nly if φ = lim φ k , { φ k } ⊂ Or b G ( f ). This is equiv a le n t to the existence o f a sequence { h k } ⊂ G with φ k = h k ∗ f . The condition φ = lim( h k ∗ f ) implies that for every g ∈ G ther e exists a k ( g ) ∈ N with: φ ( g ) = h k ∗ f ( g ) for all k ≥ k ( g ). Definition 2.4. Let G , F as above. A map f : G → F is called ap erio dic if the equation b ∗ f = f implies b = e . If the equatio n b ∗ f = f holds for s ome b ∈ G we call f b -p erio dic and b is called a p erio d of f . Definition 2.5 . (LA1) Let G , F b e as ab ove. A map f : G → F is called limit ap erio dic if and only if every φ ∈ Or b G ( f ) is ap erio dic. Definition 2.6 . (LA2) Let G , F b e as ab ove. A map f : G → F will b e called limit ap erio dic if f or every g ∈ G \ { e } there exists a finite set S ⊆ G , S = S ( g ), such that for every h ∈ G ther e is a c ∈ S with f ( hc ) 6 = f ( hg c ). Prop ositio n 2.7. These two definitions ar e e quivalent for finitely gener ate d gr oups. Pro of. Supp ose that f sa tisfies the ( L A 2) prop erty but not the ( LA 1). Then there exists a φ ∈ Or b G ( f ) such that φ has p erio d g 6 = e . Then g − 1 is a lso a per io d. Let { h k } k ∈ N ∈ G such that φ = lim h k ∗ f . Cho ose the set S for tha t g . Since S is finite we also have that g · S is finite. F rom the fact tha t φ = lim h k ∗ f , there exists a n n ∈ N suc h that for all k ≥ n and for all x ∈ S S g · S we have φ ( x ) = h k ∗ f ( x ). W e apply LA2 for f with g and h − 1 n to obtain c ∈ S such that f ( h − 1 n c ) 6 = f ( h − 1 n g c ). This contradicts with the fact that g − 1 is a per io d for φ : φ ( c ) = ( h n ∗ f )( c ) = f ( h − 1 n c ) 6 = f ( h − 1 n g c ) = ( h n ∗ f )( g c ) = φ ( g c ) = ( g − 1 ∗ φ )( c ) . Let’s s uppo se now that f satisfies the (LA1) but no t the (LA2). Then there exists a g ∈ G such that for every finite subset S of G there exists a n h ∈ G with the prop erty: f ( hc ) = f ( hg c ) ON LIMIT APERIODIC G-SETS 3 for a ll c ∈ S . Fix that g ∈ G . T ake S 1 = { c ∈ G : d ( c, e ) ≤ 1 } The distance mentioned is the o ne induced b y the word metr ic in the Cayley g raph o f G . Since G is f.g. | S 1 | < ∞ , so, there exists an h 1 ∈ G with f ( h 1 c ) = f ( h 1 g c ) for all c ∈ S 1 . T ake S 2 = { c ∈ G : d ( c, e ) ≤ 2 } . Again | S 2 | < ∞ . Then there exis ts an h 2 ∈ G such that f ( h 2 c ) = f ( h 2 g c ) for all c ∈ S 2 . Contin ue for any k ∈ N . Thu s we obtain a s equence { h k } k ∈ N ∈ G . T a king a subs e quence we ma y assume that there is a limit: φ = lim k →∞ h − 1 k ∗ f . The cla im is that φ is p erio dic with p er io d g . Co nsider an arbitr ary x ∈ G . Name k 1 = d ( x, e ), then x ∈ S k for a ll k ≥ k 1 . Also sinc e φ is the limit o f h − 1 k ∗ f there exists a k 2 ∈ N s uch that for all k ≥ k 2 : φ ( x ) = ( h − 1 k ∗ f )( x ) Finally since φ is the limit of h k ∗ f ther e exists a k 3 ∈ N such that for all k ≥ k 3 : φ ( g x ) = ( h − 1 k ∗ f )( g x ) Thu s, for k ≥ max { k 1 , k 2 , k 3 } we hav e: φ ( x ) = ( h − 1 k ∗ f )( x ) = f ( h k x ) = f ( h k g x ) = ( h − 1 k ∗ f )( g x ) = φ ( gx ) = ( g − 1 ∗ φ )( x ) . Since x was taken a rbitrarily , we hav e that φ has g − 1 as a p erio d. This is a contradiction s ince φ b elongs to the Or b G ( f ) and f has the (LA1) prop erty .  Definition 2.8. A finitely generated group G will be called limit ap erio dic if it admits a limit a per io dic co loring f : G → F with a finite set o f c o lors F . Remark 2. 9. The definition of limit ap erio dic gr oups c an e asily b e extende d to any gr oup and not only finitely gener ate d ones. Both t he pr op erty (LA1) and (LA2) apply to gr oups without the f.g. hyp othesis. Their e qu ivalenc e though dep ends on the fact that the gr oup is finitely gener ate d. F or u s a gr oup (not ne c essarily finitely gener ate d) wil l b e limit ap erio dic if it satisfies the ( LA1) pr op erty. W e re call the notion o f uniform ap erio dicit y from [1]. Befor e we introduce that notion lets establish s ome no ta tion: Notation. Let Γ be the Cayley g raph o f a group G and d be the a sso ciated metr ic. W e denote the displac ement of g at h with: d g ( h ) = d ( g h, h ) With B r ( h ) we denote the ba ll of radius r with center h . Finally k g k , the norm of g , is the dista nce b etw ee n g and e namely: k g k = d ( g , e ) Definition 2.1 0 . Let G be a finitely g enerated gro up. A map f : G → F where F is a finite set (of co lors) will b e called uniformly ap erio dic (UA) if there exis ts a constant λ > 0 s uch that for every element g ∈ G \ { e } and every h ∈ G , there exists b ∈ B λd g ( h ) ( h ) w ith f ( g b ) 6 = f ( b ). Definition 2.11. A finitely generated group is called u niformly ap erio dic if ther e exists a n F a nd a f as ab ov e, so that f : G → F is unifor mly ap er io dic. 4 THANOS GENTIMIS Prop ositio n 2 .12. If f : G → F is uniformly ap erio dic then f is li mit ap erio dic. Pro of. W e sho w that f satisfies LA2. Let g ∈ G \ { e } a nd h ∈ G arbitrary chosen. Define S = B λ k g k ( e ) to b e the ball with center e and ra dius λ k g k . Clearly since G is finitely g enerated, S is finite. Assume that there exists an h ∈ G such that for every c ∈ S w e have f ( hc ) = f ( hg c ). Denote a = hg h − 1 . W e apply the UA condition for f with a and h to obta in b in B λd a ( h ) ( h ) w ith f ( ab ) 6 = f ( b ). Since b ∈ B λd a ( h ) ( h ) we have that: d ( b, h ) ≤ λd a ( h ) = λd ( hgh − 1 h, h ) = λd ( hg, h ) = λd ( g , e ) = λ k g k where the third equalit y comes fro m the fact that the metric is left in v ariant. Notice that: d ( h − 1 b, e ) = d ( h − 1 b, h − 1 h ) = d ( b, h ) Thu s d ( h − 1 b, e ) ≤ λ k g k . This implies tha t c = h − 1 b b elongs to S . So f ( b ) = f ( h ( h − 1 b )) = f ( hc ) = f ( hg c ) = f ( hg h − 1 b ) = f ( ab ) which is clea rly a contradiction.  3. G-Sets A nd Limit A periodicity The no tion o f limit ape rio dicity can b e generalized in the ca se of G -Sets. Namely let X b e a space and supp ose tha t G acts on X giving it the structure o f a G -set. W e will use the notation g x = g ( x ) for g ∈ G and x ∈ X . Fix a finite set F , which we ca n consider ag ain as color s. Denote by F X the set of all maps from X to F . Then F X can b ecome a G -set under the following a ction: ( g ∗ f )( x ) = f ( g − 1 x ) for a ll x ∈ X , g ∈ G and f ∈ F X . Also denote b y: F ix G ( X ) = { g ∈ G : g · x = x, ∀ x ∈ X } the kernel of the action. W e naturally get the following definitions : Definition 3 .1. Let X b e a G -set and let f ∈ F X . W e call f limit ap erio dic if and only if for every φ ∈ Or b G ( f ) we hav e that φ is ap erio dic meaning that if a ∗ φ = φ , then a ∈ F ix ( X ). Thu s we get the definition o f limit ap er io dic G -sets: Definition 3.2 . Let X b e a G -set. If there exists a finite set F a nd a map f : X → F such that f is limit ap erio dic we say tha t X is a limit ap erio dic G -set . Remark 3.3. If we c onsider a gr oup G acting on itself with left mu ltiplic ation then G is limit ap erio dic as a G -set if and only if G is limit ap erio dic as a gr oup, b e c ause under that action F ix ( G ) = { e } . Let X be a G - set and let G x = S tab G ( x ) = { g ∈ G | g x = x } deno te the stabilizer of x ∈ X . ON LIMIT APERIODIC G-SETS 5 Theorem 3.4. L et X b e limit ap erio dic G -set and supp ose that G acts tr ansitively on X . Fix x ∈ X su ch that X = Or b G ( x ) . If S tab G ( x ) is a limit ap erio dic gr oup for some x ∈ X t hen G is a limit ap erio dic gr oup. Pro of. Let φ : X → F 1 be a limit aper io dic ma p for X and let ψ : G x → F 2 be a limit ap erio dic map for G x . W e k now that there exists a bijection π b etw een the s et o f left cosets G/ G x and the orbit Or b G ( x ). Fix a set of repr esentativ es in G na mely { a i : i ∈ I } for the quotients G/G x . Then π ( a j G x ) = a j x . Define f = ( f 1 , f 2 ) : G → F 1 × F 2 by f 1 ( g ) = φ ( g x ) and f 2 ( g ) = ψ ( a − 1 j g ) where g ∈ a j G x . W e will prove that this f is a limit aperio dic map. Suppo se that this is false. Then there exists a map f ∈ Or b G ( f ) and an ele men t a ∈ G such that f has a as a per io d, i.e., ( a ∗ f ) = f for a ll x ∈ X . Let f = lim h k ∗ f where h k ∈ G Case 1) Suppo se a / ∈ G x . Consider the limit φ = lim k h k ∗ φ. Note that we can alwa ys c ho ose a subsequence of h k such that the limit exists. F or conv enience w e keep the same indices for the subsequence . Clear ly , φ ∈ O rb G ( φ ). Given g ∈ G , there exis ts k 0 ∈ N s uch that for all k ≥ k 0 we have: φ ( g x ) = ( h k ∗ φ )( g x ) Also there exists a k 1 ∈ N s uch that for all k ≥ k 1 we get: f ( g ) = ( h k ∗ f )( g ). Let k 2 = max { k 0 , k 1 } then for all k ≥ k 2 we have f ( g ) = f ( h − 1 k g ). Therefor e, f 1 ( g ) = f 1 ( h − 1 k g ) = φ ( h − 1 k g x ) = ( h k ∗ φ )( g x ) = φ ( g x ) . F ollowing the previous pro of a nd replacing g with a − 1 g we find a k 3 ∈ N such that for a ll k ≥ k 3 we have: f 1 ( a − 1 g ) = φ ( a − 1 g x ) Since f 1 has p er io d a we hav e ( a ∗ f )( g ) = f ( g ). Hence, ( a ∗ φ )( g x ) = φ ( a − 1 g x ) = f 1 ( a − 1 g ) = ( a ∗ f 1 )( g ) = f 1 ( g ) = φ ( a − 1 g x ) . Since g is arbitra ry and G a cts on X tra nsitively we get that for every y ∈ X , ( a ∗ φ )( y ) = φ ( y ). Since φ is limit ap erio dic we hav e that a ∈ F ix ( X ). But: F ix ( X ) = \ s ∈ S S tab G ( s ) Thu s, a ∈ F i x ( X ) ⊆ S tab G ( x ) = G x contradiction. Case 2) Suppo se that a ∈ G x . Let { h k } b e a sequence of elements o f G such that h − 1 k belo ngs to the coset a k G x . Th us δ k = h k a k belo ngs to G x . T aking a subsequence we may as sume that there are the limits ψ = lim k δ k ∗ ψ and f = lim k h k ∗ f . Notice that ψ ∈ Or b G x ( ψ ). Let h ∈ G x . Then there ex ists a k 0 ∈ N s uc h that for a ll k ≥ k 0 we have ψ ( h ) = ( δ k ∗ ψ )( h ) 6 THANOS GENTIMIS Also ther e exists a k 1 ∈ N s uch that for all k ≥ k 1 we g e t: f | G x ( h ) = ( h k ∗ f )( h ) Then for all k ≥ max { k 0 , k 1 } we hav e: f | G x ( h ) = f ( h − 1 k h ). Hence, ( f | G x ) 2 ( h ) = ψ ( a − 1 k h − 1 k h ) = ψ (( h k a k ) − 1 h ) = (( h k a k ) ∗ ψ )( h ) = ( δ k ∗ ψ )( h ) = ψ ( h ) . Notice now that: a ∗ ψ = a ∗ f | G x = f | G x = ψ . Thu s ψ is per io dic. Contradiction since ψ is limit aper io dic. This concludes the pro of.  The following is o bvious. Lemma 3.5. If X is a G -sp ac e and H is any gr oup acting on X such that the action of H factors thr ough t he action of G . Then if X is a limit ap erio dic G -sp ac e then it is also a limit ap erio dic H -sp ac e. Corollary 3.6. If G and H ar e limit ap erio dic gr oups and 1 → G τ − → E π − → H → 1 is a short exact se quenc e then E is also limit ap erio dic. Pro of. Ob viously E acts transitively o n E /G = H with left mult iplication. By Lemma 3 .5 H is a limit aperio dic E -space. Note tha t S tab E ( e H ) = G is limit ap erio dic. If we apply Theor e m 3.4 we get the coro llary .  Obviously we obtain the fo llowing Corollary 3.7. If G is limit ap erio dic and H is limit ap erio dic t hen G × H is limit ap erio dic. Corollary 3. 8. If H is a limit ap erio dic gr oup and θ : H → H is a gr oup auto- morphism t hen the HNN extension ⋆ θ H is limit ap erio dic. Pro of. W e know that if H = < S | T > where S is a s et o f gener ators and T is a set of r elations then G = ⋆ θ H admits the following presentation ⋆ θ H = < S, t | T ∪ { t − 1 xt = θ ( x ) , x ∈ S } > . Note that G a cts transitively on the set G/H of all left cosets o f H . Note that G/H = { t i H | i ∈ Z } ∼ = Z . Thus, G acts on Z by trans la tions with S tab G ( { H } ) = H [2]. W e color the Z with a Morse-Thue sequence as in [1], i.e. φ : X → { 0 , 1 } with φ ( t i H ) = m ( | i | ) wher e m : N → { 0 , 1 } is the Mor se-Thue sequenc e [3],[4]. As it shown in [1] this map is limit ap erio dic with r esp ect. By Lemma 3.5 it is a limit ap erio dic G -set. Theo rem 3 .4 co mpletes the pro o f.  In order to pr ov e the fact that the fre e pro duct of limit ap er io dic groups is limit ap erio dic we need the following notions . Let A and B b e tw o g roups. W e will construct a set X such that the free pro duct G = A ⋆ B acts o n X fr eely and transitively and X is a limit ap erio dic G -space . Let T 0 be the Bass-Ser r e tree asso ciated with A ⋆ B . W e r ecall that the vertices o f T 0 are left cose ts G/ A ∪ G/B and the vertices o f the type xA , xB and only th em for m an e dge [ xA, xB ] in T 0 . Thu s the edg es of T 0 are in the bij ection with G . Note that G acts o n T 0 by left m ultiplication. Let T b e the ba rycentric sub divisio n of T 0 and let X b e the set o f the baryc en ters (of edges). W e will identify the tree T with the s et of its vertices. Then the group G acts by iso metries on T yielding three or bits o n the vertices ON LIMIT APERIODIC G-SETS 7 X = O r b G ( e ), G/ A and G/B . W e r egard T as a ro oted tree with the r o ot e . Let k x k = d T ( x, e ) denote the distance to the ro ot. Lemma 3.9. L et A,B b e limit ap erio dic gr oups and let G = A ⋆ B , X , T define d as ab ove. Then T is a limit ap erio dic G -set. Pro of. Let π : G → A with π ( w ) = π ( a 1 b 1 a 2 b 2 ...a n b n ) = a 1 a 2 ...a n and θ : G → B with θ ( w ) = θ ( a 1 b 1 a 2 b 2 ...a n b n ) = b 1 b 2 ...b n . Clear ly bo th π and θ are group homo mo rphisms. Let a lso f A : A → F A be the limit a per io dic map for the g r oup A a nd f B : B → F B be the limit aperio dic map for group B . Also let ν : Z → { 0 , 1 , 2 } b e the v ar iation o f the Morse -Thue seq uence which has no words W W (see for example [1]). Also fix e to b e the vertex r epresenting t he identit y element in T . Then cons ider a colo ring of T as follows: f : T → { 0 , 1 , 2 } × { 0 , 1 , 2 } × ( F A [ { α } ) × ( F B [ { β } ) = F where f := ( f 0 , f 1 , f 2 , f 3 ) with: f 0 ( x ) = ν ( k x k ), f 1 ( x ) = k x k mod 3, f 2 ( x ) = f A ( π ( x )) if x ∈ X and f 2 ( x ) = α if x ∈ T − X . Fina lly let f 3 ( x ) = f B ( θ ( x )) if x ∈ X and f 3 ( x ) = β if x ∈ T − X . The group G acts on the space of colo rings F T as fo llows: ( g ∗ f )( x ) = (( g ∗ f 0 )( x ) , ( g ∗ f 1 )( x ) , ( π ( g ) ∗ f 2 )( x ) , ( θ ( g ) ∗ f 3 )( x ) = ( f 0 ( g − 1 x ) , f 1 ( g − 1 x ) , f 2 ( π − 1 ( g ) x ) , f 3 ( θ − 1 ( g ) x )) . Suppo se that f is not a limit ap erio dic map. Then there exists a c oloring ψ = ( ψ 0 , ψ 1 , ψ 2 , ψ 3 ) such that ψ ∈ Or b G ( f ) and ψ has a p erio d b ∈ G \ F ix ( T ). Let ψ = lim g k ∗ f . Then ψ A = lim π ( g k ) ∗ f A has p erio d π ( b ). Indeed, for every x ∈ A ⊂ G ∼ = X for large eno ugh k , ( π ( g k ) ∗ f A )( x ) = f A ( π ( g − 1 k x )) = f 2 ( g − 1 k x ) = ( g k ∗ f 2 )( x ) = ( g k ∗ f 2 )( bx ) = f 2 ( g − 1 k bx ) = f A ( π ( g − 1 k bx )) = ( π ( g k ) ∗ f A )( π ( b ) x ) . Similarly ψ B = lim θ ( g k ) ∗ f B has a p erio d θ ( b ). Thus, π ( b ) = e A and θ ( b ) = e B . Notice that ( ψ 0 , ψ 1 ) ∈ Or b G ( f 0 , f 1 ). Denote ξ = ( ψ 0 , ψ 1 ) and φ = ( f 0 , f 1 ). Then ξ is a coloring of a simplicial tree ( T ) on whic h G acts by isometries. Moreov er ξ ∈ Or b G ( φ ). F rom prop osition 4, pa ge 31 8 in [1] we hav e that b ∗ ξ 6 = ξ for all b ∈ G with unbo unded orbit { b k x 0 | k ∈ Z } . T his clearly implies that b ∗ ψ 6 = ψ for every b ∈ G with unbounded orbit. On the other hand ψ has p erio d b and thus w e hav e { b k x 0 | k ∈ N } is b o unded. This implies that b fix e s a p oint in T . Call that po int x 1 . Since the a ction of G on X is free, x 1 / ∈ X . Thus, x 1 ∈ G/ A or x 1 ∈ G/B . Assume the later, x 1 = wB for some w inG . Since b fixes w B , b = wb ′ w − 1 for some b ′ ∈ B \ { e B } . Then θ ( b ) = θ ( w ) b ′ θ ( w ) − 1 6 = e B . Contradiction.  Lemma 3.10. L et G = A ⋆ B , T , X , f , F as a b ove. Then X is a limit ap erio dic G -set. Pro of. Note that in the ro o ted tree T e very vertex x 6 = e has a unique pre - decessor deno ted pred ( x ). W e define f ′ : X → F × F as f ′ ( x ) = ( f | X , ˆ f ) where ˆ f ( x ) = f ( pr ed ( x )). W e show that f ′ is limit ap erio dic . Suppo se that f ′ is not limit ap erio dic. The n there exists a sequence { g k } ∈ G s.t. ψ ′ = lim k g k ∗ f ′ 8 THANOS GENTIMIS and b ∈ F ix G ( X ) with b ∗ ψ ′ = ψ ′ . W e may ass ume that there is the limit ψ = lim g k ∗ f . In view o f Le mma 3.9 it s uffice s to show that ψ is b -p erio dic. It is b -p erio dic on X , so it suffice to chec k that it is b -p erio dic on T \ X . Let z ∈ T \ X . W e chec k that ψ ( bz ) = ψ ( z ). Since the r o ot e lies in X , we ma y a ssume that z 6 = e . Let x 0 = pr ed ( z ) and let x 1 be such that z = pred ( x 1 ). W e note that x 1 is no t unique. So we fix one. Note that x 0 , x 1 ∈ X . There is k 0 such that for k ≥ k 0 ψ ′ ( x i ) = ( g k ∗ f ′ )( x i ) , ψ ′ ( bx i ) = ( g k ∗ f ′ )( bx i ) , i = 0 , 1 and ψ ( z ) = ( g k ∗ f )( z ) , ψ ( b z ) = ( g k ∗ f )( bz ) . Fix k ≥ k 0 . Since G a cts o n T by iso metries the distance fro m g − 1 k z to g − 1 k x i , i = 0 , 1 equals 1. There are three p ossibilities: (1) g − 1 k x 0 < g − 1 k z < g − 1 k x 1 , (2) g − 1 k x 1 < g − 1 k z < g − 1 k x 0 , and (3) g − 1 k z < g − 1 k x i , i = 0 , 1 . W e apply g − 1 k bg k . In view o f the fact that f 1 ( g − 1 k x i ) = f 1 ( g − 1 k bx i ), i = 0 , 1 we obtain g − 1 k bx 0 < g − 1 k bz < g − 1 k bx 1 in the case (1) and g − 1 k bx 1 < g − 1 k bz < g − 1 k bx 0 , in the case (2). Then in the case (1) f ′ ( g − 1 k bx 1 ) = ( g k ∗ f ′ )( bx 1 ) = ψ ′ ( bx 1 ) = ψ ′ ( x 1 ) = ( g k ∗ f ′ )( x 1 ) = f ′ ( g − 1 k x 1 ) . Hence f 0 ( g − 1 k bx 1 ) = f 0 ( g − 1 k x 1 ). Therefore f ( g − 1 k bz ) = f ( g − 1 k z ) . Thu s, ψ ( bz ) = ( g k ∗ f )( bz ) = f ( g − 1 k bz ) = f ( g − 1 k z ) = ( g k ∗ f )( z ) = ψ ( z ) . In the case (2) we consider x 0 instead o f x 1 . In the case (3 ) g − 1 k bz is the predecessor of either g − 1 k bx 0 or g − 1 k bx 1 (or both). Assume the first. Then from the b -p erio dicity of ψ ′ it follo ws that f 0 ( g − 1 k bx 0 ) = f 0 ( g − 1 k x 0 ). Since f 0 ( g − 1 k bx 0 ) = f ( pre d ( g − 1 k bx 0 )) and f 0 ( g − 1 k x 0 ) = f ( pr ed ( g − 1 k x 0 )), we o bta in ψ ( bz ) = f ( g − 1 k bz ) = f ( g − 1 k z ) = ψ ( z ).  ON LIMIT APERIODIC G-SETS 9 Theorem 3. 11. L et A , B b e limit ap erio dic gr oups. Then G = A ⋆ B is a limit ap erio dic gr oup. Pro of. By Lemma 3.10 X is a limit ap erio dic G -set. Note that G a c ts on X as ab ov e transitiv ely , and S tab G ( x 0 ) = { e } is a limit aperio dic group. By Theorem 3.4 we have that G is a limit ap erio dic group.  W e finish this pap er with an example of a G -set which is not limit ap erio dic . Prop ositio n 3.12 . Consider the automorphism gr oup of the inte gers Aut ( Z ) . L et s : Z → Z with s ( n ) = n + 1 and t : Z → Z wi th t (0) = 1 , t (1) = 0 and t ( n ) = n if n 6 = 1 and n 6 = 0 . L et S = < s, t > then N is not a li mit ap erio dic S -set . Pro of. Supp ose that N w as a limit aper io dic S -set. Then let F be a set o f co lors with | F | < ∞ and a map f : N → F s uch that f is a limit ap erio dic map under the action of S . Since | F | < ∞ a nd | N | = ∞ there exists a t leas t one a ∈ F s uch that infinitely many a n ∈ N hav e f ( a n ) = a . Cho ose a strictly incr easing sequence in N such that f ( a n ) = a for all n ∈ N . Consider the following elements in S : h 1 = (1 , a 1 ) h 2 = (1 , a 1 )(2 , a 2 ) . . . h n = (1 , a 1 )(2 , a 2 ) . . . ( n, a n ) . . . where ( i , a i ) is the trans po sition that takes i to a i . With s a nd t we can construct all the transp ositions. Thus all a i belo ng to S . Clea r ly if n ≥ k , n, k ∈ N we hav e that h n ( k ) = a k . Consider now the seq uence { h − 1 n ∗ f } and take a co n verging subsequence. F or convenience in notatio n let us keep the same indice s for the subsequence. Thus if: ψ = lim n →∞ h − 1 n ∗ f we ha ve that ψ ∈ Or b S ∞ ( f ). Thus ψ has to b e ap erio dic. The claim is that ψ ( k ) = a for all k ∈ N and th us ψ is clea rly p erio dic whic h will lead to a contradiction. This is easy t o see since le t k ∈ N . Then for th at k there exists an n 1 such that for all n ≥ n 1 we hav e that ψ ( k ) = ( h − 1 n ∗ f )( k ). Thus fo r n = max { k , n 1 } we have that: ψ ( k ) = ( h − 1 n ∗ f )( k ) = f ( h n k )) = f ( a k ) = a. This co ncludes the pro o f. 1 References [1] A.Dr anishniko v and V. Sc hroeder Ap erio dic Colorings And Tilings Of Coxeter Gr oups Groups Geom. Dyn. 1(2007 ), 311-328. [2] J.P . Serre T r e es , Springer- V erl ag(2003 ), ISBN 3540442375 [3] H.M . Morse and G.A. Hedlund Unending c hess, symb olic dynamics and a pr oblem in semi- gr oups Duke Math. J. 11 (194 4), 1-7. [4] A. Thue Ub er unend liche Zeic henr eihen Christiania Vidensk. Selsk. Skr. No 7 (1906) , p. 22. Ma thematic s Dep ar tm ent, University of Florida, Gainesville , USA E-mail addr ess : genti mis@math .ufl.edu 1 When this paper w as finalized we receiv ed information about a pr eprint on limit ap erio dic colorings of groups by Gao, Jac kson and Seaw ard found in h ttp://www.cas.un t.edu/ ∼ sgao/pub/pub.h tml