Overview First Proof General Proof Group Colorings and Shift Equivalence Relations S. Gao, S. Jackson*, B. Seward Department of Mathematics University of North Texas AMS-ASL Special Session on Logic and Dynamical Systems January, 2009 Washington, DC S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Overview Recall the definition of a 2-coloring of a countable group G . Definition c : G → { 0 , 1 } is a 2 -coloring if ∀ s ∈ G ∃ T ∈ G <ω ∀ g ∈ G ∃ t ∈ T ( c ( gt ) � = c ( gst )) . This definition was formulated independently by Pestov (c.f. paper of Glasner and Uspenski). S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Significance of the definition. Let E be the shift equivalence relation on X = 2 G , given by the action of G : g · x ( h ) = x ( g − 1 h ) . Let F denote the free part of this space, that is, x ∈ F iff ∀ g � = 1 ( g · x � = x ) . 1. Coloring property gives a marker compactness property. (MCP) Let S 0 ⊇ S 1 ⊇ S 2 ⊇ · · · be relatively closed complete sections of F . Then � n S n � = ∅ . 2. Coloring property is equivalent to a free orbit closure. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Main Result Theorem Every countable group G has the 2 -coloring property. ◮ First proof works for abelian, solvable groups. ◮ Second proof works for general groups. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Main Result Theorem Every countable group G has the 2 -coloring property. ◮ First proof works for abelian, solvable groups. ◮ Second proof works for general groups. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Other Combinatorial Reformulations Other natural descriptive properties have combinatorial reformulations in terms of the group G . Definition Colorings c 1 , c 2 of G are orthogonal ( c 1 ⊥ c 2 ) if ∃ T ∈ G <ω ∀ g 1 , g 2 ∈ G ∃ t ∈ T ( c 1 ( g 1 t ) � = c 2 ( g 2 t )) . Fact If x , y ∈ F, then x ⊥ y iff [ x ] ∩ [ y ] = ∅ . S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Definition A coloring c is minimal if ∀ S ∈ G <ω ∃ T ∈ G <ω ∀ g ∈ G ∃ t ∈ T ∀ s ∈ S ( c ( s ) = c ( gts )) . Fact x ∈ F is minimal iff [ x ] is minimal (i.e., for every y ∈ [ y ] we have [ x ] = [ y ] ). S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Significance Overview Main Result First Proof Other Reformulations General Proof Extensions Extension of Result Theorem For every countable group G there is a perfect set of pairwise orthogonal, minimal orbits in F. In fact, we get more..... S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary First consider the simplest case of G = Z . The following is not the argument that works in general, but has applications. We define two sequences a i , b i from 2 <ω . We will have lh( a i ) = lh( b i ). Can take b i = 1 − a i . Each a i +1 , (and b i +1 ) is a concatenation of a i ’s and b i ’s. (May assume lh( a i ) > i + 1). Let a i +1 = a i b i a i a i b i , b i +1 = b i a i b i b i a i . S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary a i +1 a i b i a i a i b i Let x be any concatenations of a i +1 ’s and b i +1 ’s. Then for s = i + 1, can take T = { 0 , 1 , . . . , 2lh( a i +1 ) } . to verify the 2-coloring property for this s . To get a coloring, take any x such that for each i , x is a concatenation of a i ’s and b i ’s. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary Easily modify to get a perfect set of pairwise orthogonal 2-colorings. For example, for w ∈ 2 ω define x ( w ) as above but using a i +1 = a i b i a i a i c i b i where � a i if x ( i ) = 0 c i = b i if x ( i ) = 1 a i +1 a i b i a i a i c i b i S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary Each x ( w ) has the following marker identification property : (MIP) There is a finite A ⊆ Z such that for any k ∈ Z , whether k · x is the start of an a i or b i is determined by k · x ↾ A . In fact A depends only on i , not on w . If i is least such that w 1 ( i ) � = w 2 ( i ), then x ( w 1 ) ⊥ x ( w 2 ) follows from the marker identification property, using roughly A i + | a i | . S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary Extending To Other G We can extend this method to show the following. Theorem Suppose Z � G. Then G has the 2 -coloring property. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary proof sketch. Let x 1 Z , x 2 Z , . . . be the cosets of Z in G = { g 1 , g 2 , . . . } . If g / ∈ Z , then g induces a fixed-point-free permutation π g on the cosets. We use the algorithm above to color each coset x i Z with a 2-coloring c i . At step i , if g i ∈ Z then we define the a i , b i for each coset as above. If g i / ∈ Z then consider π i = π g i . On each orbit of π i , if π i ( x Z ) = y Z , then define the a i , b i for x Z and for y Z such that the colorings will be orthogonal, and by a fixed set A i (not depending on x and y ). To see this works, for s ∈ G take cases as to whether s ∈ Z . If s ∈ Z , the 2-coloring property is satisfied by the argument that ∈ Z , then for g ∈ x j Z , gs ∈ x k Z each c n is a 2-coloring. If s = g i / for some j � = k , and the set A i witnesses the 2-coloring property for g and gs (by the orthogonality of c j and c k ). S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary These methods give: Corollary Every abelian, and in fact, every solvable group has the 2-coloring property. This method can be used to show more, for example: Theorem Let Z � G. Then the set of 2 -colorings of G is Π 0 3 -complete. In summary, these methods show: ◮ Every abelian or solvable group has the 2-coloring property. ◮ If Z � G or S � G where S is infinite solvable, then G has the 2-coloring property. ◮ Show directly the free group F n has the 2-coloring property. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary These methods give: Corollary Every abelian, and in fact, every solvable group has the 2-coloring property. This method can be used to show more, for example: Theorem Let Z � G. Then the set of 2 -colorings of G is Π 0 3 -complete. In summary, these methods show: ◮ Every abelian or solvable group has the 2-coloring property. ◮ If Z � G or S � G where S is infinite solvable, then G has the 2-coloring property. ◮ Show directly the free group F n has the 2-coloring property. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Case G=Z Overview A modification First Proof Other G General Proof Summary These methods give: Corollary Every abelian, and in fact, every solvable group has the 2-coloring property. This method can be used to show more, for example: Theorem Let Z � G. Then the set of 2 -colorings of G is Π 0 3 -complete. In summary, these methods show: ◮ Every abelian or solvable group has the 2-coloring property. ◮ If Z � G or S � G where S is infinite solvable, then G has the 2-coloring property. ◮ Show directly the free group F n has the 2-coloring property. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Overview Ideas First Proof Marker Regions General Proof The coloring Two main ideas: 1. Get reasonable marker regions for general groups. 2. Exploit polynomial versus exponential growth. S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
Overview Ideas First Proof Marker Regions General Proof The coloring Marker Regions Question What kind of marker regions can we get for general groups? Say a group G has regular markers if there are E 0 ⊆ E 1 ⊆ E 2 ⊆ · · · , each E i an equivalence relation on G with finite classes each of which is a translate by a fixed set A i ⊆ G , and such that � i E i = G × G . Question Which groups admit regular markers? S. Gao, S. Jackson*, B. Seward Group Colorings and Shift Equivalence Relations
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