Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a...
Transcript of Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a...
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Rank rigidity of CAT(0) groups
Dan P. Guralnik (Penn), Eric L. Swenson (BYU)
February 16, 2012
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 2: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/2.jpg)
The CAT(0) inequality
A complete geodesic metric space (X, d) is CAT(0), if allgeodesic triangles Mxyz are thinner than their Euclideancomparison triangles:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 3: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/3.jpg)
CAT(0) – Why should I care?
Properties:Convexity of the metric
convex analysis can be done here!Uniqueness of geodesicsNearest point projections to closed convex subspacesContractibilityUniversal cover of locally CAT(0) is CAT(0), e.g.:
- a 2-dim’l square complex with vertex links of girth≥ 4;- more generally a cubical complex all whose links are flags.
a starategy for constructing a K(G, 1)A compact set has a circumcenter
have a grip on compact subgroups of Isom (X)
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 4: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/4.jpg)
CAT(0) Groups
By a CAT(0) group we mean a group G, together with a proper,co-compact isometric (geometric) action on a CAT(0) space X.
Properties:A finite order element fixes a point in X (elliptic);An element of infinite order acts as a translation on ageodesic line in X (hyperbolic);There are only finitely many elliptic conjugacy classes;No infinite subgroup of G is purely elliptic. (Swenson)
Theorem (Flat Torus theorem)If H < G is a free abelian subgroup of rank d, then H stabilizesan isometrically embedded d-flat F ⊂ X, on which it actsco-compactly by translations.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 5: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/5.jpg)
Examples⟨a, b
∣∣− ⟩× Z acts geometrically on the product of a 4-regulartree with the real line:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 6: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/6.jpg)
Examples
Other examples:Lattice in Isom(Hn): becomes a CAT(0) group uponexcising a maximal disjoint family of precisely invarianthoroballs;Coxeter groups acting on Davis-Moussong complexes;Direct products of free groups;More generally, right-angled Artin groups;Fundamental groups of piecewise-NPC complexes withlarge links.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Visual Boundary
Two geodesic rays γ, γ′ : [0,∞)→ X are asymptotic, ifd(γ(t), γ′(t)) is bounded.
The visual boundary ∂X of a CAT(0) space X is the set ofasymptoticity classes of geodesic rays in X.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 8: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/8.jpg)
Visual Boundary
Two geodesic rays γ, γ′ : [0,∞)→ X are asymptotic, ifd(γ(t), γ′(t)) is bounded.
The visual boundary ∂X of a CAT(0) space X is the set ofasymptoticity classes of geodesic rays in X.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 9: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/9.jpg)
Visual Boundary
Two geodesic rays γ, γ′ : [0,∞)→ X are asymptotic, ifd(γ(t), γ′(t)) is bounded.
The visual boundary ∂X of a CAT(0) space X is the set ofasymptoticity classes of geodesic rays in X.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 10: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/10.jpg)
Visual Boundary
Two geodesic rays γ, γ′ : [0,∞)→ X are asymptotic, ifd(γ(t), γ′(t)) is bounded.
The visual boundary ∂X of a CAT(0) space X is the set ofasymptoticity classes of geodesic rays in X.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 11: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/11.jpg)
Visual Boundary
Two geodesic rays γ, γ′ : [0,∞)→ X are asymptotic, ifd(γ(t), γ′(t)) is bounded.
The visual boundary ∂X of a CAT(0) space X is the set ofasymptoticity classes of geodesic rays in X.
TheoremLet x ∈ X. Then every asymptoticity class in ∂X contains aunique representative emanating from x.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Why boundaries?
Boundaries were born to answer coarse geometricquestions, e.g.:
Which subgroups of G stabilize large-scale features?
. . . for example, extend the idea (and role) of parabolicsubgroups encountered in the classical groups; radial vs.tangential convergence.
If X is so-and-so, what is G?
. . . Mostow rigidity utilizes the conformal structure on theideal sphere for classifying G up to conjugacy according tothe homotopy type of G\H3.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Cone boundary ∂∞X vs. Tits boundary ∂TX
Two ideas for topologizing ∂X:Two boundary points are close if. . .
. . . a pair of representative rays fellow-travels for a longtime; (Cone topology, ∂∞X – use projections to balls)
. . . I can stare at both of them at the same time, no matterwhere I stand. (Tits metric, ∂TX – use angles)
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 14: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/14.jpg)
Cone boundary ∂∞X vs. Tits boundary ∂TX
Two ideas for topologizing ∂X:Two boundary points are close if. . .
. . . a pair of representative rays fellow-travels for a longtime; (Cone topology, ∂∞X – use projections to balls)
. . . I can stare at both of them at the same time, no matterwhere I stand. (Tits metric, ∂TX – use angles)
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 15: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/15.jpg)
Cone boundary ∂∞X vs. Tits boundary ∂TX
Two ideas for topologizing ∂X:Two boundary points are close if. . .
. . . a pair of representative rays fellow-travels for a longtime; (Cone topology, ∂∞X – use projections to balls)
. . . I can stare at both of them at the same time, no matterwhere I stand. (Tits metric, ∂TX – use angles)
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 16: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/16.jpg)
Cone boundary ∂∞X vs. Tits boundary ∂TX
Example: G = F2 × Z y T4 × R
∂ (T4 × R) = ∂T4 ∗ ∂R = {Cantor set} ∗ {±∞}
∂∞X is coarser than ∂TX!!
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 17: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/17.jpg)
Cone boundary ∂∞X vs. Tits boundary ∂TX
More generally, we have good news:∂TX is a complete CAT(1) space (Kleiner-Leeb)
∂(X × Y) = ∂X ∗ ∂Y for both the Cone and Titsboundaries (Berestovskij)
The Tits metric is lower semi-continuous on ∂∞X × ∂∞X.
The following are equivalent: (Gromov?)- G is Gromov-hyperbolic,- ∂TX is discrete,- X contains no 2-flat.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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The Bad News
∂TX is not locally compact (F2 × Z)G one-ended but ∂TX not connected (Croke-Kleiner)G determines neither ∂∞X nor ∂TX (Croke-Kleiner)Many join-irreducible examples of ∂∞X not locallyconnected (Mihalik-Ruane)∂∞X not 1-connected though G is 1-connected at infinity
(Mihalik-Tschantz)If there is a round Sd ⊂ ∂TX, is there a periodic Ed+1 ⊂ X?
(Gromov, Wise)∂TX is connected iff diam∂TX ≤ 3π
2 .(Ballmann-Buyalo, Swenson-Papasoglu)
What does it mean for ∂TX to have diameter≤ π?
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Rank One
Let X be a proper CAT(0) space.
Rank oneA rank one geodesic in X is a geodesic line not bounding a flathalf-plane. A rank one isometry is a hyperbolic isometryg ∈ Isom (X) having a rank one axis. A group G < Isom (X)has rank one if it contains a rank one isometry (otherwise G hashigher rank).
Origin: rank one Lie groups and discrete subgroupsthereof;More generally: hyperbolic and relatively-hyperbolicgroups;Typical behaviour: Convergence dynamics, mimickingcompactness properties of univalent analytic mappings.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Rank One and Convergence Dynamics
Discrete Convergence (Gehring-Martin)Every infinite F ⊂ G contains a sequence gn converging on∂∞X to a constant map uniformly on compacts in ∂∞X r {pt}.
Morally (or loosely) speaking,G of higher rank ⇔ flats abound in X ⇒ DCG fails
How? – e.g., if g ∈ G has an axis bounding a flat half-plane F,then gn cannot collapse ∂F to g(∞).
Some properties of rank one groups:Many non-abelian free subgroupsMore ‘interesting’ boundariesBetter chances for splittings over ‘nice’ subgroups
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Compressiblity
To study higher rank groups, we introduce:
Compressible pairs (Swenson-G.)A pair p, q ∈ ∂X is G-compressible if there are gn ∈ G such thatgnp→ p∞, gnq→ q∞ but dT(p∞, q∞) < dT(p, q).A ⊂ ∂X is incompressible if contains no compressible pair.
Examples:Rank one ⇒ No non-degenerate incompressible setsX = Em ⇒ entire boundary is G-incompressibleMore generally, ∂TX compact ⇒ ∂TX is G-incompressible
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Example: G = F2 × F2 is compressibleAction of gn = un × vn with 1 6= u, v ∈ F2 causes massivecompressions away from the repelling points (ξ, η arbitrary):
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Example: G = F2 × F2 is compressibleAction of gn = un × vn with 1 6= u, v ∈ F2 causes massivecompressions away from the repelling points (ξ, η arbitrary):
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 24: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/24.jpg)
Example: G = F2 × F2 is compressibleAction of gn = un × vn with 1 6= u, v ∈ F2 causes massivecompressions away from the repelling points (ξ, η arbitrary):
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 25: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/25.jpg)
Example: G = F2 × F2 is compressibleAction of gn = un × vn with 1 6= u, v ∈ F2 causes massivecompressions away from the repelling points (ξ, η arbitrary):
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 26: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/26.jpg)
Example: G = F2 × F2 is compressibleAction of gn = un × vn with 1 6= u, v ∈ F2 causes massivecompressions away from the repelling points (ξ, η arbitrary):
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 27: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/27.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 28: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/28.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 29: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/29.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 30: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/30.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 31: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/31.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 32: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/32.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 33: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/33.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 34: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/34.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 35: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/35.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 36: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/36.jpg)
Example: G = F2 × F2 is compressibleAny subsequence with unk · u−∞ and vnk · v−∞ converging givesrise to a well-defined limiting ‘folding’ operator on ∂TX:
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 37: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/37.jpg)
Example: G = F2 × F2 is compressibleQuestion: How much of this can be retained without priorknowledge about group/space structure?
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Goal: the Rank-Rigidity Conjectures
Suppose G y X is a CAT(0) group.
Conjecture: Closing Lemma (Ballman-Buyalo)If diam∂TX > π, then G has rank one.
The best known bound is 3π2 , due to Swenson and Papasoglu.
Conjecture: Rank-rigidity (Ballman-Buyalo)If diam∂TX = π and X is irreducible, then X is either asymmetric space or a Euclidean building.
Known for:Riemannian manifolds (Ballman)Cell complexes of low dimensions (Ballman and Brin)Cubings (Caprace and Sageev)
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 39: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/39.jpg)
A sample of our resultsTheorem (G.-Swenson)Let G y X be a CAT(0) group of higher rank, and let d denotethe geometric dimension of ∂TX. Then
diam∂TX ≤ 2π − arccos(− 1
d + 1
).
Theorem (G.-Swenson)Let G y X be a CAT(0) group of higher rank. TFAE:
1 G is virtually-Abelian;2 X contains a virtually G-invariant coarsely dense flat;3 G stabilizes a non-degenerate maximal incompressible
subset of ∂X.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Approach through the boundary ∂TX
Most promising results in the direction of rank rigidity are:Leeb: If X is geodesically-complete and ∂TX is ajoin-irreducible spherical building then X is a symmetricspace or Euclidean building;Lytchak: If ∂TX is geodesically-complete and contains aproper closed subspace closed under taking antipodes, then∂TX is a spherical building.
Question: Assume G has higher rank. How to use G y X forobtaining a classification of its possible boundaries?
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 41: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/41.jpg)
Some known structural results
Let Z be a finite-dimensional complete CAT(1) space:Lytchak: If Z is geodesically-complete then
Z = Sn︸︷︷︸sphere
∗Z1 ∗ · · · ∗ Zk︸ ︷︷ ︸irred. buildings
∗ Y1 ∗ · · · ∗ Yl︸ ︷︷ ︸irred. none of the above
This decomposition is unique.Swenson: There always is a decomposition
Z = S(Z)︸︷︷︸sphere
∗ E(Z)︸︷︷︸no sphere factor
Moreover, S(Z) is the set of suspension points of Zand the decomposition is unique.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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π-Convergence
Let G be a group of isometries of a CAT(0) space X.
π-Convergence (Swenson-Papasoglu)
For every ε ∈ [0, π], every infinite F ⊂ G contains a sequencegn converging on ∂∞X into a Tits ball BT(p, ε) uniformly oncompacts in ∂∞X r BT(n, π − ε).
D. P. Guralnik Rank Rigidity of CAT(0) groups
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π-Convergence
Let G be a group of isometries of a CAT(0) space X.
π-Convergence (Swenson-Papasoglu)
For every ε ∈ [0, π], every infinite F ⊂ G contains a sequencegn converging on ∂∞X into a Tits ball BT(p, ε) uniformly oncompacts in ∂∞X r BT(n, π − ε).
D. P. Guralnik Rank Rigidity of CAT(0) groups
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π-Convergence
Let G be a group of isometries of a CAT(0) space X.
π-Convergence (Swenson-Papasoglu)
For every ε ∈ [0, π], every infinite F ⊂ G contains a sequencegn converging on ∂∞X into a Tits ball BT(p, ε) uniformly oncompacts in ∂∞X r BT(n, π − ε).
Special attention on the words ‘converging’ and ‘into’:1 No actual limiting map ∂∞X r BT(n, π − ε)→ BT(p, ε).2 Varying limits can be constructed, but are –
choice-dependent, andrestricted to Tits-compact sets.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Our contribution: use the Ellis semi-group
Instead of limits of the form limn→∞
gnp, work with G-ultra-limits
ωp = limg→ω
gp, computed in the cone compactification X̂ of X.
G discrete and countable, soSpace of ultra-filters on G with Tychonoff topologycoincides with βG, soG y X̂ extends to a semi-group action βG y X̂, andThe inversion map extends to a continuous involutionω 7→ Sω (antipode).Caveat:
1 ω y ∂∞X need not be continuous,2 S(ων) 6= (Sν)(Sω).
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 46: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/46.jpg)
Our contribution: use the Ellis semi-group
Instead of limits of the form limn→∞
gnp, work with G-ultra-limits
ωp = limg→ω
gp, computed in the cone compactification X̂ of X.
G discrete and countable, so
Space of ultra-filters on G with Tychonoff topologycoincides with βG, soG y X̂ extends to a semi-group action βG y X̂, andThe inversion map extends to a continuous involutionω 7→ Sω (antipode).Caveat:
1 ω y ∂∞X need not be continuous,2 S(ων) 6= (Sν)(Sω).
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 47: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/47.jpg)
Our contribution: use the Ellis semi-group
Instead of limits of the form limn→∞
gnp, work with G-ultra-limits
ωp = limg→ω
gp, computed in the cone compactification X̂ of X.
G discrete and countable, soSpace of ultra-filters on G with Tychonoff topologycoincides with βG, so
G y X̂ extends to a semi-group action βG y X̂, andThe inversion map extends to a continuous involutionω 7→ Sω (antipode).Caveat:
1 ω y ∂∞X need not be continuous,2 S(ων) 6= (Sν)(Sω).
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 48: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/48.jpg)
Our contribution: use the Ellis semi-group
Instead of limits of the form limn→∞
gnp, work with G-ultra-limits
ωp = limg→ω
gp, computed in the cone compactification X̂ of X.
G discrete and countable, soSpace of ultra-filters on G with Tychonoff topologycoincides with βG, soG y X̂ extends to a semi-group action βG y X̂, and
The inversion map extends to a continuous involutionω 7→ Sω (antipode).Caveat:
1 ω y ∂∞X need not be continuous,2 S(ων) 6= (Sν)(Sω).
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 49: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/49.jpg)
Our contribution: use the Ellis semi-group
Instead of limits of the form limn→∞
gnp, work with G-ultra-limits
ωp = limg→ω
gp, computed in the cone compactification X̂ of X.
G discrete and countable, soSpace of ultra-filters on G with Tychonoff topologycoincides with βG, soG y X̂ extends to a semi-group action βG y X̂, andThe inversion map extends to a continuous involutionω 7→ Sω (antipode).
Caveat:1 ω y ∂∞X need not be continuous,2 S(ων) 6= (Sν)(Sω).
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 50: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/50.jpg)
Our contribution: use the Ellis semi-group
Instead of limits of the form limn→∞
gnp, work with G-ultra-limits
ωp = limg→ω
gp, computed in the cone compactification X̂ of X.
G discrete and countable, soSpace of ultra-filters on G with Tychonoff topologycoincides with βG, soG y X̂ extends to a semi-group action βG y X̂, andThe inversion map extends to a continuous involutionω 7→ Sω (antipode).Caveat:
1 ω y ∂∞X need not be continuous,2 S(ων) 6= (Sν)(Sω).
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 51: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/51.jpg)
Ultra-filter Miracles
Suppose ω ∈ βG is non-principal.
Contraction property. ω is a Lip-1 operator on ∂TX.Sink and Source. ω is constant on X, so define –
ω(∞) = ωX , ω(−∞) = (Sω)X .
π-Convergence. for all p ∈ ∂X, ε ∈ [0, π]:
dT(p, ω(−∞)) ≥ π − ε ⇒ dT(ωp, ω(∞)) ≤ ε .
Compression. A incompressible iff ω restricts to anisometry on A, for all ω ∈ βG.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Ultra-filter Miracles
Suppose ω ∈ βG is non-principal.Contraction property. ω is a Lip-1 operator on ∂TX.
Sink and Source. ω is constant on X, so define –
ω(∞) = ωX , ω(−∞) = (Sω)X .
π-Convergence. for all p ∈ ∂X, ε ∈ [0, π]:
dT(p, ω(−∞)) ≥ π − ε ⇒ dT(ωp, ω(∞)) ≤ ε .
Compression. A incompressible iff ω restricts to anisometry on A, for all ω ∈ βG.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 53: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/53.jpg)
Ultra-filter Miracles
Suppose ω ∈ βG is non-principal.Contraction property. ω is a Lip-1 operator on ∂TX.Sink and Source. ω is constant on X, so define –
ω(∞) = ωX , ω(−∞) = (Sω)X .
π-Convergence. for all p ∈ ∂X, ε ∈ [0, π]:
dT(p, ω(−∞)) ≥ π − ε ⇒ dT(ωp, ω(∞)) ≤ ε .
Compression. A incompressible iff ω restricts to anisometry on A, for all ω ∈ βG.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 54: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/54.jpg)
Ultra-filter Miracles
Suppose ω ∈ βG is non-principal.Contraction property. ω is a Lip-1 operator on ∂TX.Sink and Source. ω is constant on X, so define –
ω(∞) = ωX , ω(−∞) = (Sω)X .
π-Convergence. for all p ∈ ∂X, ε ∈ [0, π]:
dT(p, ω(−∞)) ≥ π − ε ⇒ dT(ωp, ω(∞)) ≤ ε .
Also...
Compression. A incompressible iff ω restricts to anisometry on A, for all ω ∈ βG.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 55: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/55.jpg)
Ultra-filter Miracles
Suppose ω ∈ βG is non-principal.Contraction property. ω is a Lip-1 operator on ∂TX.Sink and Source. ω is constant on X, so define –
ω(∞) = ωX , ω(−∞) = (Sω)X .
π-Convergence. for all p ∈ ∂X, ε ∈ [0, π]:
dT(p, ω(−∞)) ≥ π − ε ⇒ dT(ωp, ω(∞)) ≤ ε .
Also...Compression. A incompressible iff ω restricts to anisometry on A, for all ω ∈ βG.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 56: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/56.jpg)
Ultra-filter Miracles
Suppose ω ∈ βG is non-principal.Contraction property. ω is a Lip-1 operator on ∂TX.Sink and Source. ω is constant on X, so define –
ω(∞) = ωX , ω(−∞) = (Sω)X .
π-Convergence. for all p ∈ ∂X, ε ∈ [0, π]:
dT(p, ω(−∞)) ≥ π − ε ⇒ dT(ωp, ω(∞)) ≤ ε .
Also...Compression. A incompressible iff ω restricts to anisometry on A, for all ω ∈ βG.
Most importantly: Can appeal to compactness of βG.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 57: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/57.jpg)
Our main tool: Folding and Total Folding
Folding Lemma (Swenson-G.)Let d be the geometric dimension of ∂TX. Then for every(d + 1)-flat F there exist ω0 ∈ βG and a (d + 1)-flat F0 suchthat
1 ω0 maps ∂F isometrically onto S0 = ∂F0, and2 ω0 maps ∂X onto the sphere S0.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Our main tool: Folding and Total Folding
Folding Lemma (Swenson-G.)Let d be the geometric dimension of ∂TX. Then for every(d + 1)-flat F there exist ω0 ∈ βG and a (d + 1)-flat F0 suchthat
1 ω0 maps ∂F isometrically onto S0 = ∂F0, and2 ω0 maps ∂X onto the sphere S0.
Corollaries: For S0 as above,every minimal closed invariant subset of ∂∞X intersects S0;S0 contains an isometric copy of any incompressible subsetA ⊂ ∂X;Every maximal incompresible subset of ∂X is isometric toa compact π-convex subset of a round sphere.
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Our main tool: Folding and Total Folding
Total Folding (Swenson-G.)There exists ν0 ∈ βG such that
ν0∂X is a maximal incompressible subset of maximalvolume (MVI),ν2
0 = ν0 in βG, and ν0∂X ⊂ S0.Moreover, any two MVI’s are isometric and (geometrically)interiorly-disjoint.
Remarks:Rank one implies ∂X is compressible. Converse?Does higher rank imply S0 is covered by incompressibles?
A positive answer implies diam∂TX = π
Is ∂X covered by MVI’s? Through Lytchak, this would imply rank rigidity!
D. P. Guralnik Rank Rigidity of CAT(0) groups
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Endspiel: let’s try to prove something
Theorem (G.-Swenson)Let G y X be a CAT(0) group of higher rank. TFAE:
1 G is virtually-Abelian;2 X contains a virtually G-invariant coarsely dense flat;3 G stabilizes a non-degenerate maximal incompressible
subset of ∂X.
The plan to prove (3) ⇒ (1):Prove that ∂TX and ∂∞X coincide with the round sphere;Hit this on the head with Shalom’s QI characterization ofvirtually-Abelian groups.
But first we need to find a candidate sphere living inside ∂X.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 61: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/61.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.
A pole is an element of a dipole.Let P be the set of all poles.
!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).! S(∂TX) is incompressible, and therefore P = S(∂TX).
Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 62: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/62.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.A pole is an element of a dipole.
Let P be the set of all poles.!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).! S(∂TX) is incompressible, and therefore P = S(∂TX).
Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 63: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/63.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.A pole is an element of a dipole.Let P be the set of all poles.
!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).! S(∂TX) is incompressible, and therefore P = S(∂TX).
Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 64: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/64.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.A pole is an element of a dipole.Let P be the set of all poles.
!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).
! S(∂TX) is incompressible, and therefore P = S(∂TX).Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 65: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/65.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.A pole is an element of a dipole.Let P be the set of all poles.
!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).! S(∂TX) is incompressible, and therefore P = S(∂TX).
Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 66: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/66.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.A pole is an element of a dipole.Let P be the set of all poles.
!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).! S(∂TX) is incompressible, and therefore P = S(∂TX).
Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.
In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 67: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/67.jpg)
The sphere of poles
We now apply Swenson’s decomposition of ∂TX:A dipole is an incompressible pair p, q with dT(p, q) = π.A pole is an element of a dipole.Let P be the set of all poles.
!! Poles are suspension points of ∂TX (so P ⊂ S(∂TX)).! S(∂TX) is incompressible, and therefore P = S(∂TX).
Every (non-degenerate) max incompressible contains P:A ∪ P is incompressible, so P ⊆ A by maximality.In fact, A is the spherical join of P with a compact convexspherical polytope.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 68: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/68.jpg)
Characterizing virtually-Abelian groups
Main direction: if G stabilizes a max incompressible set A, thenG is virtually-Abelian.
Write A = P ∗ B using Swenson’s decomposition, andprove B must be empty (This is the main step where grouptheory is involved).
Thus, P is the only max incompressible, but that impliesE(∂TX) is empty, by G-invariance, and we are done.The main step: unless B is empty, G has a fixed point in B;now use Ruane’s result to virtually split G as a directproduct with a Z factor, allowing an induction on dimP.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 69: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/69.jpg)
Characterizing virtually-Abelian groups
Main direction: if G stabilizes a max incompressible set A, thenG is virtually-Abelian.
Write A = P ∗ B using Swenson’s decomposition, andprove B must be empty (This is the main step where grouptheory is involved).Thus, P is the only max incompressible, but that impliesE(∂TX) is empty, by G-invariance, and we are done.
The main step: unless B is empty, G has a fixed point in B;now use Ruane’s result to virtually split G as a directproduct with a Z factor, allowing an induction on dimP.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 70: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/70.jpg)
Characterizing virtually-Abelian groups
Main direction: if G stabilizes a max incompressible set A, thenG is virtually-Abelian.
Write A = P ∗ B using Swenson’s decomposition, andprove B must be empty (This is the main step where grouptheory is involved).Thus, P is the only max incompressible, but that impliesE(∂TX) is empty, by G-invariance, and we are done.The main step: unless B is empty, G has a fixed point in B;now use Ruane’s result to virtually split G as a directproduct with a Z factor, allowing an induction on dimP.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 71: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/71.jpg)
Characterizing virtually-Abelian groups
Main direction: if G stabilizes a max incompressible set A, thenG is virtually-Abelian.
Write A = P ∗ B using Swenson’s decomposition, andprove B must be empty (This is the main step where grouptheory is involved).Thus, P is the only max incompressible, but that impliesE(∂TX) is empty, by G-invariance, and we are done.The main step: unless B is empty, G has a fixed point in B;now use Ruane’s result to virtually split G as a directproduct with a Z factor, allowing an induction on dimP.
THE END, THANK YOU!
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 72: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/72.jpg)
What Makes Folding Work?
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 73: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/73.jpg)
What Makes Folding Work?
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 74: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/74.jpg)
What Makes Folding Work?
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 75: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/75.jpg)
What Makes Folding Work?
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 76: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/76.jpg)
What Makes Folding Work?
Properties of pulling (Swenson-G.)Suppose ω ∈ βG pulls from a point n ∈ ∂X. Then:
1 if F is a flat with n ∈ ∂F, then ω maps ∂F isometricallyonto the boundary of a flat;
2 if dT(n, a) ≤ π, then ω restricts to an isometry on [n, a];3 if dT(n, a) ≥ π, then ωa = ω(∞). Thus,4 ω maps ∂X into the geodesic suspension of ωn and ω(∞),
preserving boundaries of flats through n.
D. P. Guralnik Rank Rigidity of CAT(0) groups
![Page 77: Rank rigidity of CAT(0) groups - kodlab.seas.upenn.edu fileCAT(0) Groups By a CAT(0) group we mean a group G, together with aproper, co-compact isometric (geometric) actionon a CAT(0)](https://reader030.fdocuments.us/reader030/viewer/2022041302/5e128466f8657d3c343cc3f8/html5/thumbnails/77.jpg)
What Makes Folding Work?
D. P. Guralnik Rank Rigidity of CAT(0) groups