Generalized Expanders
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1 Bounded Jerry Kaminker University of California, Davis Lucas Sabalka Binghamton University 28 March, 2009
2 Outline Bounded Bounded 4
3 Bounded Coarse Embeddings Definition (coarse embedding) A family (X i ) i I coarsely embeds into a metric space Y if there exists embeddings (F i : X i Y ) which are uniformly coarse: there exist increasing unbounded ρ, ρ + with, for all i I and x, y X i, ρ (d(x, y)) d(f i (x), F i (y)) ρ + (d(x, y)). impose conditions on asymptotic geometry - boundaries into linear spaces is strong - for f.g. gps, implies Novikov into Hilbert spaces related to amenability, exactness, Gromov s a-t-menability/haagerup, finite asymptotic dimension, coarse Baum-Connes, Novikov,...
4 Bounded Coarse Embeddings Definition (coarse embedding) A family (X i ) i I coarsely embeds into a metric space Y if there exists embeddings (F i : X i Y ) which are uniformly coarse: there exist increasing unbounded ρ, ρ + with, for all i I and x, y X i, ρ (d(x, y)) d(f i (x), F i (y)) ρ + (d(x, y)). impose conditions on asymptotic geometry - boundaries into linear spaces is strong - for f.g. gps, implies Novikov into Hilbert spaces related to amenability, exactness, Gromov s a-t-menability/haagerup, finite asymptotic dimension, coarse Baum-Connes, Novikov,...
5 Bounded Madoff s Theorem Theorem (Gromov) If a graph has a Ponzi flow, it is not amenable.
6 Bounded Madoff s Theorem Theorem (Gromov) If a graph has a Ponzi flow, it is not amenable.
7 Bounded Madoff s Theorem Theorem (Gromov) If a graph has a Ponzi flow, it is not amenable.
8 Bounded Madoff s Theorem Theorem (Gromov) If a graph has a Ponzi flow, it is not amenable.
9 Bounded Getting the most out of your telephone network A classical expander (Bassalygo, Pinsker) is a family of sparse, highly connected graphs. Usually defined in terms of uniformly bounded isoperimetric number (Cheeger number, spectral gap) h(g) := S min 1 S G S 2 Amazingly useful in applications (networks (AT&T), error correction, pseudorandomness, Markov chains, Monte Carlo algorithms, etc)
10 Bounded Getting the most out of your telephone network A classical expander (Bassalygo, Pinsker) is a family of sparse, highly connected graphs. Usually defined in terms of uniformly bounded isoperimetric number (Cheeger number, spectral gap) h(g) := S min 1 S G S 2 Amazingly useful in applications (networks (AT&T), error correction, pseudorandomness, Markov chains, Monte Carlo algorithms, etc)
11 Bounded Getting the most out of your telephone network A classical expander (Bassalygo, Pinsker) is a family of sparse, highly connected graphs. Usually defined in terms of uniformly bounded isoperimetric number (Cheeger number, spectral gap) h(g) := S min 1 S G S 2 Amazingly useful in applications (networks (AT&T), error correction, pseudorandomness, Markov chains, Monte Carlo algorithms, etc)
12 Bounded One of the Definitions Definition (Jerrum, Sinclair) A sequence of finite connected graphs (X n ) is a classical expander if: 1 there s a uniform bound k on the degrees of vertices, 2 X n, 3 C > 0 such that, for all n and all f n l 2 (X n ), 1 X n 2 x,y X n f (x) f (y) 2 C X n x,y X n x,y adjacent f (x) f (y) 2.
13 Results Bounded The 3-regular graphs G p, p prime, with vertex set Z p and edge set {(x, x ± 1), (x, x 1 ) x Z p } (uses Selberg 3/16 from Number Theory) Theorem (Margulis; see Lubotsky) Families of Schreier graphs of finite-index subgroups of groups with Property T form an expander. Theorem (Gromov) Let H denote a separable infinite-dimensional Hilbert space. A classical expander (X n ) does not coarsely embed into H.
14 Results Bounded The 3-regular graphs G p, p prime, with vertex set Z p and edge set {(x, x ± 1), (x, x 1 ) x Z p } (uses Selberg 3/16 from Number Theory) Theorem (Margulis; see Lubotsky) Families of Schreier graphs of finite-index subgroups of groups with Property T form an expander. Theorem (Gromov) Let H denote a separable infinite-dimensional Hilbert space. A classical expander (X n ) does not coarsely embed into H.
15 Results Bounded The 3-regular graphs G p, p prime, with vertex set Z p and edge set {(x, x ± 1), (x, x 1 ) x Z p } (uses Selberg 3/16 from Number Theory) Theorem (Margulis; see Lubotsky) Families of Schreier graphs of finite-index subgroups of groups with Property T form an expander. Theorem (Gromov) Let H denote a separable infinite-dimensional Hilbert space. A classical expander (X n ) does not coarsely embed into H.
16 Bounded Preliminaries For r > 0, metric space X, the r-diagonal complement is Ω r (X) := {(x, y) X 2 d(x, y) r}. Let (X, µ) be a sequence X n with measures µ n defined on Xn 2, such that: µ n either zero or probability measure, µ n a probability measure infinitely often, There is r n > 0 such that r n, µ n supported on Ω rn (X n ). This is a measured family. Measured family (X, µ) has Poincaré inequality for metric space Z if, for all 1-Lipschitz map f : X Z : Var µn (f ) := d(f (a), f (b)) 2 µ n (a, b) K 2. (a,b) X 2 n
17 Bounded Preliminaries For r > 0, metric space X, the r-diagonal complement is Ω r (X) := {(x, y) X 2 d(x, y) r}. Let (X, µ) be a sequence X n with measures µ n defined on Xn 2, such that: µ n either zero or probability measure, µ n a probability measure infinitely often, There is r n > 0 such that r n, µ n supported on Ω rn (X n ). This is a measured family. Measured family (X, µ) has Poincaré inequality for metric space Z if, for all 1-Lipschitz map f : X Z : Var µn (f ) := d(f (a), f (b)) 2 µ n (a, b) K 2. (a,b) X 2 n
18 Bounded Preliminaries For r > 0, metric space X, the r-diagonal complement is Ω r (X) := {(x, y) X 2 d(x, y) r}. Let (X, µ) be a sequence X n with measures µ n defined on Xn 2, such that: µ n either zero or probability measure, µ n a probability measure infinitely often, There is r n > 0 such that r n, µ n supported on Ω rn (X n ). This is a measured family. Measured family (X, µ) has Poincaré inequality for metric space Z if, for all 1-Lipschitz map f : X Z : Var µn (f ) := d(f (a), f (b)) 2 µ n (a, b) K 2. (a,b) X 2 n
19 Definition Bounded Definition (Ostrovskii, Tessera) For any family of metric spaces C, (X, µ) is a C-expander, or generalized expander, if (X, µ) has a Poincaré inequality with uniform constant K for every Z C. (slightly modified from Tessera: does not reference space obstructing; only need subsequences)
20 Results Bounded Theorem (Ostrovskii, Tessera) A classical expander is a {H}-expander. Theorem (Tessera, (Ostrovskii)) Let C be a class of metric spaces satisfying some conditions. A C-expander does not coarsely embed into any Z C. A metric space X does not coarsely embed into any element Z C if and only if some C-expander coarsely embeds into X. : associated sheaf is p-admissible. Satisfied by: separable Hilbert, L p, CAT (0),...
21 Results Bounded Theorem (Ostrovskii, Tessera) A classical expander is a {H}-expander. Theorem (Tessera, (Ostrovskii)) Let C be a class of metric spaces satisfying some conditions. A C-expander does not coarsely embed into any Z C. A metric space X does not coarsely embed into any element Z C if and only if some C-expander coarsely embeds into X. : associated sheaf is p-admissible. Satisfied by: separable Hilbert, L p, CAT (0),...
22 Results Bounded Theorem (Ostrovskii, Tessera) A classical expander is a {H}-expander. Theorem (Tessera, (Ostrovskii)) Let C be a class of metric spaces satisfying some conditions. A C-expander does not coarsely embed into any Z C. A metric space X does not coarsely embed into any element Z C if and only if some C-expander coarsely embeds into X. : associated sheaf is p-admissible. Satisfied by: separable Hilbert, L p, CAT (0),...
23 Motivation Bounded For {X n, µ n }, define: and Theorem α n := 1 X n 2 min (a,b) supp(µn) (µ n (a, b)) β n := ( r 2 n X n 2 ) Ω rn (X n ) X n 2. For ν n the uniform measure on X 2 n, Z a space where X has a Poincaré inequality, and 1-Lipschitz F n : X n Z, Var νn (F n ) α n K 2 + β n. Idea: Split up Var νn (F n ) over Ω n and Ω c n:
24 Motivation Bounded For {X n, µ n }, define: and Theorem α n := 1 X n 2 min (a,b) supp(µn) (µ n (a, b)) β n := ( r 2 n X n 2 ) Ω rn (X n ) X n 2. For ν n the uniform measure on X 2 n, Z a space where X has a Poincaré inequality, and 1-Lipschitz F n : X n Z, Var νn (F n ) α n K 2 + β n. Idea: Split up Var νn (F n ) over Ω n and Ω c n:
25 Motivation Bounded For {X n, µ n }, define: and Theorem α n := 1 X n 2 min (a,b) supp(µn) (µ n (a, b)) β n := ( r 2 n X n 2 ) Ω rn (X n ) X n 2. For ν n the uniform measure on X 2 n, Z a space where X has a Poincaré inequality, and 1-Lipschitz F n : X n Z, Var νn (F n ) α n K 2 + β n. Idea: Split up Var νn (F n ) over Ω n and Ω c n:
26 Motivation Bounded For {X n, µ n }, define: and Theorem α n := 1 X n 2 min (a,b) supp(µn) (µ n (a, b)) β n := ( r 2 n X n 2 ) Ω rn (X n ) X n 2. For ν n the uniform measure on X 2 n, Z a space where X has a Poincaré inequality, and 1-Lipschitz F n : X n Z, Var νn (F n ) α n K 2 + β n. Idea: Split up Var νn (F n ) over Ω n and Ω c n:
27 Definition Bounded Definition Define α := sup n α n and β := sup n β n. A generalized expander is bounded if α, β <. All classical expanders are bounded (α n, β n 0) Not all generalized expanders are bounded?
28 Definition Bounded Definition Define α := sup n α n and β := sup n β n. A generalized expander is bounded if α, β <. All classical expanders are bounded (α n, β n 0) Not all generalized expanders are bounded?
29 Definition Bounded Definition Define α := sup n α n and β := sup n β n. A generalized expander is bounded if α, β <. All classical expanders are bounded (α n, β n 0) Not all generalized expanders are bounded?
30 Bounded Coarse Embeddings Lemma If X is bounded and has a Poincaré inequality for Z, there exists k and k so for any n and 1-Lipschitz F n : X n Z, Var νn (F n ) αk 2 + β + k = k. Theorem Let M be a Hadamard manifold, and let X be a bounded Hilbert-expander. Then X does not coarsely embed into M. Proof similar to standard argument for classical expanders.
31 Bounded Coarse Embeddings Lemma If X is bounded and has a Poincaré inequality for Z, there exists k and k so for any n and 1-Lipschitz F n : X n Z, Var νn (F n ) αk 2 + β + k = k. Theorem Let M be a Hadamard manifold, and let X be a bounded Hilbert-expander. Then X does not coarsely embed into M. Proof similar to standard argument for classical expanders.
32 Bounded Coarse Embeddings Lemma If X is bounded and has a Poincaré inequality for Z, there exists k and k so for any n and 1-Lipschitz F n : X n Z, Var νn (F n ) αk 2 + β + k = k. Theorem Let M be a Hadamard manifold, and let X be a bounded Hilbert-expander. Then X does not coarsely embed into M. Proof similar to standard argument for classical expanders.
33 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
34 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
35 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
36 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
37 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
38 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
39 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
40 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
41 Bounded Modifications Play with definition. make supp(µ n ) = Ω? assume coarsely connected? assume nested? All obstruction theorems still hold Analyze "Moduli space" of expanders Good notion of equivalence of expanders? Good topology or metric on "Moduli space" of expanders? Examples of non-classical generalized expanders? Examples of non-bounded classical expanders?
42 Bounded Boundaries What can be said about "boundaries" of expanders? Classical expanders obstruct: all cohomology Lipschitz (of Connes, Gromov, Moscovici). "Claim": So do bounded generalized expanders Measure-theoretic H uf of Block, Weinberger? Understand how classes in cohomology of a group might satisfy Novikov, but not come from a "boundary".
43 Bounded Boundaries What can be said about "boundaries" of expanders? Classical expanders obstruct: all cohomology Lipschitz (of Connes, Gromov, Moscovici). "Claim": So do bounded generalized expanders Measure-theoretic H uf of Block, Weinberger? Understand how classes in cohomology of a group might satisfy Novikov, but not come from a "boundary".
44 Bounded Boundaries What can be said about "boundaries" of expanders? Classical expanders obstruct: all cohomology Lipschitz (of Connes, Gromov, Moscovici). "Claim": So do bounded generalized expanders Measure-theoretic H uf of Block, Weinberger? Understand how classes in cohomology of a group might satisfy Novikov, but not come from a "boundary".
45 Bounded Boundaries What can be said about "boundaries" of expanders? Classical expanders obstruct: all cohomology Lipschitz (of Connes, Gromov, Moscovici). "Claim": So do bounded generalized expanders Measure-theoretic H uf of Block, Weinberger? Understand how classes in cohomology of a group might satisfy Novikov, but not come from a "boundary".
46 Bounded Boundaries What can be said about "boundaries" of expanders? Classical expanders obstruct: all cohomology Lipschitz (of Connes, Gromov, Moscovici). "Claim": So do bounded generalized expanders Measure-theoretic H uf of Block, Weinberger? Understand how classes in cohomology of a group might satisfy Novikov, but not come from a "boundary".
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