The shell lemma Applications Some properties of group actions on zero-dimensional spaces Colin D. Reid University of Newcastle, Australia Trees, dynamics and locally compact groups, HHU Düsseldorf, June 2018 Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications Let X be a locally compact Hausdorff topological space and write CO ( X ) for the set of compact open subsets of X . Suppose that X is zero-dimensional , meaning that CO ( X ) forms a base for the topology. Let S ⊆ Homeo ( X ) , such that id X ∈ S , S = S − 1 and { sU | s ∈ S } is finite for every U ∈ CO ( X ) . Let S n be the set of products of at most n elements of S , and let G = S ∞ = � S � . Fix some U ∈ CO ( X ) . Write U 0 = U ; for n ∈ ( 0 , + ∞ ] , U n = � g ∈ S n gU and U − n = � g ∈ S n gU . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications Let X be a locally compact Hausdorff topological space and write CO ( X ) for the set of compact open subsets of X . Suppose that X is zero-dimensional , meaning that CO ( X ) forms a base for the topology. Let S ⊆ Homeo ( X ) , such that id X ∈ S , S = S − 1 and { sU | s ∈ S } is finite for every U ∈ CO ( X ) . Let S n be the set of products of at most n elements of S , and let G = S ∞ = � S � . Fix some U ∈ CO ( X ) . Write U 0 = U ; for n ∈ ( 0 , + ∞ ] , U n = � g ∈ S n gU and U − n = � g ∈ S n gU . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications Let X be a locally compact Hausdorff topological space and write CO ( X ) for the set of compact open subsets of X . Suppose that X is zero-dimensional , meaning that CO ( X ) forms a base for the topology. Let S ⊆ Homeo ( X ) , such that id X ∈ S , S = S − 1 and { sU | s ∈ S } is finite for every U ∈ CO ( X ) . Let S n be the set of products of at most n elements of S , and let G = S ∞ = � S � . Fix some U ∈ CO ( X ) . Write U 0 = U ; for n ∈ ( 0 , + ∞ ] , U n = � g ∈ S n gU and U − n = � g ∈ S n gU . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications The space U −∞ = � g ∈ G gU is open in X (so locally compact) and G -invariant. We think of U −∞ as partitioned into a ‘core’ U + ∞ (compact, but not necessarily open) and a sequence of ‘shells’ W n := U n \ U n + 1 indexed by the integers (each of which is compact and open). Lemma (i) There exist a , b ∈ [ −∞ , + ∞ ] with a ≤ 0 ≤ b such that U a = U −∞ , U b = U ∞ and W m is nonempty exactly when m ∈ [ a , b ) . (ii) Every G -orbit intersecting U n \ U + ∞ also intersects W m for all m ∈ [ a , n ] . (iii) There is a G -orbit Gx that intersects all of the nonempty shells. Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications The space U −∞ = � g ∈ G gU is open in X (so locally compact) and G -invariant. We think of U −∞ as partitioned into a ‘core’ U + ∞ (compact, but not necessarily open) and a sequence of ‘shells’ W n := U n \ U n + 1 indexed by the integers (each of which is compact and open). Lemma (i) There exist a , b ∈ [ −∞ , + ∞ ] with a ≤ 0 ≤ b such that U a = U −∞ , U b = U ∞ and W m is nonempty exactly when m ∈ [ a , b ) . (ii) Every G -orbit intersecting U n \ U + ∞ also intersects W m for all m ∈ [ a , n ] . (iii) There is a G -orbit Gx that intersects all of the nonempty shells. Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications The space U −∞ = � g ∈ G gU is open in X (so locally compact) and G -invariant. We think of U −∞ as partitioned into a ‘core’ U + ∞ (compact, but not necessarily open) and a sequence of ‘shells’ W n := U n \ U n + 1 indexed by the integers (each of which is compact and open). Lemma (i) There exist a , b ∈ [ −∞ , + ∞ ] with a ≤ 0 ≤ b such that U a = U −∞ , U b = U ∞ and W m is nonempty exactly when m ∈ [ a , b ) . (ii) Every G -orbit intersecting U n \ U + ∞ also intersects W m for all m ∈ [ a , n ] . (iii) There is a G -orbit Gx that intersects all of the nonempty shells. Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications The space U −∞ = � g ∈ G gU is open in X (so locally compact) and G -invariant. We think of U −∞ as partitioned into a ‘core’ U + ∞ (compact, but not necessarily open) and a sequence of ‘shells’ W n := U n \ U n + 1 indexed by the integers (each of which is compact and open). Lemma (i) There exist a , b ∈ [ −∞ , + ∞ ] with a ≤ 0 ≤ b such that U a = U −∞ , U b = U ∞ and W m is nonempty exactly when m ∈ [ a , b ) . (ii) Every G -orbit intersecting U n \ U + ∞ also intersects W m for all m ∈ [ a , n ] . (iii) There is a G -orbit Gx that intersects all of the nonempty shells. Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications Proof (i) Suppose for some b ≥ 0 that W b = ∅ , i.e. U b = U b + 1 , and let m ≥ 0. Then � � U b + m = gU b = gU b + 1 = U b + m + 1 . g ∈ S m g ∈ S m Hence U b + 1 = U b + 2 = · · · = U + ∞ . The proof in the negative direction is similar. (ii) Let x ∈ U n \ U + ∞ . Then x ∈ W n ′ for some n ′ ≥ n , and hence there exists g ∈ S such that gx �∈ U n ′ (otherwise we would have x ∈ U n ′ + 1 ), but gx ∈ U n ′ − 1 (since x ∈ U n ′ ). Thus gx ∈ W n ′ − 1 . Repeat to get images of x in W m for all m ≤ n ′ . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications Proof (i) Suppose for some b ≥ 0 that W b = ∅ , i.e. U b = U b + 1 , and let m ≥ 0. Then � � U b + m = gU b = gU b + 1 = U b + m + 1 . g ∈ S m g ∈ S m Hence U b + 1 = U b + 2 = · · · = U + ∞ . The proof in the negative direction is similar. (ii) Let x ∈ U n \ U + ∞ . Then x ∈ W n ′ for some n ′ ≥ n , and hence there exists g ∈ S such that gx �∈ U n ′ (otherwise we would have x ∈ U n ′ + 1 ), but gx ∈ U n ′ − 1 (since x ∈ U n ′ ). Thus gx ∈ W n ′ − 1 . Repeat to get images of x in W m for all m ≤ n ′ . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications g ∈ S n g − 1 U n ) \ U 1 . Then P n is a compact (iii) Define P n = ( � subset of U . Let I be the set of n ≥ 0 such that W n � = ∅ . n ∈ I P n � = ∅ . Given part (ii) it is enough to show � Suppose x ∈ P n . Then ∃ g ∈ S , h ∈ S n − 1 : ghx ∈ U n , so hx ∈ U n − 1 and hence x ∈ P n − 1 . Thus ( P n ) n ∈ I is a descending sequence. Suppose � n ∈ I P n = ∅ . Then by compactness P n = ∅ for some n ∈ I , that is, g − 1 U n ⊆ U 1 for all g ∈ S n . But then U n ⊆ � g ∈ S n gU 1 = U n + 1 , so W n = ∅ , contradicting the choice of n . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications g ∈ S n g − 1 U n ) \ U 1 . Then P n is a compact (iii) Define P n = ( � subset of U . Let I be the set of n ≥ 0 such that W n � = ∅ . n ∈ I P n � = ∅ . Given part (ii) it is enough to show � Suppose x ∈ P n . Then ∃ g ∈ S , h ∈ S n − 1 : ghx ∈ U n , so hx ∈ U n − 1 and hence x ∈ P n − 1 . Thus ( P n ) n ∈ I is a descending sequence. Suppose � n ∈ I P n = ∅ . Then by compactness P n = ∅ for some n ∈ I , that is, g − 1 U n ⊆ U 1 for all g ∈ S n . But then U n ⊆ � g ∈ S n gU 1 = U n + 1 , so W n = ∅ , contradicting the choice of n . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications g ∈ S n g − 1 U n ) \ U 1 . Then P n is a compact (iii) Define P n = ( � subset of U . Let I be the set of n ≥ 0 such that W n � = ∅ . n ∈ I P n � = ∅ . Given part (ii) it is enough to show � Suppose x ∈ P n . Then ∃ g ∈ S , h ∈ S n − 1 : ghx ∈ U n , so hx ∈ U n − 1 and hence x ∈ P n − 1 . Thus ( P n ) n ∈ I is a descending sequence. Suppose � n ∈ I P n = ∅ . Then by compactness P n = ∅ for some n ∈ I , that is, g − 1 U n ⊆ U 1 for all g ∈ S n . But then U n ⊆ � g ∈ S n gU 1 = U n + 1 , so W n = ∅ , contradicting the choice of n . Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
The shell lemma Applications Alternative incarnation of (iii) (think of G = X acting by conjugation on itself, and U a vertex stabilizer): Lemma/Corollary Let Γ be a connected locally finite graph and let G be a closed vertex-transitive group of automorphisms of Γ . Then exactly one of the following holds: (i) There is a finite set v 1 , . . . , v n of vertices, such that � n i = 1 G v i = { 1 } . (ii) There is a horoball H in Γ , such that the pointwise fixator of H in G is nontrivial. Here we define a horoball to be a set of the form { v ∈ V Γ : ∃ n : d ( v , v n ) ≤ n } , where ( v n ) n ≥ 0 is a set of vertices forming a geodesic ray. Colin Reid University of Newcastle, Australia Group actions on zero-dimensional spaces
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