Groups of polynomial growth

Transcription

1 Groups of polynomial growth (after Gromov, Kleiner, and Shalom Tao) Dave Witte Morris University of Lethbridge, Alberta, Canada dave.morris Abstract. M. Gromov proved in 1981 that every group of polynomial growth has a nilpotent subgroup of finite index. This is a fundamental result in Geometric Group Theory, but Gromov s proof is too difficult to include in many standard courses on the subject. A tremendous simplification of the proof was achieved by B. Kleiner in 2010 (using methods of T. H. Colding and W. P. Minicozzi II). A further improvement by Y. Shalom and T. Tao has made the theorem truly accessible. We present the simplified proof. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

2 Geometric Group Theory G = group (e.g., Z 2 ) S = symmetric finite generating set (e.g., {(±1, 0), (0, ±1)}) x, y G: y = x? = xs 1 s 2 s k, Metric: d(x, y) = least such k. s i S, k N So G has both algebraic structure (group) and geometric structure (metric space). Basic question: How are they related? Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

3 Gromov s Thm Definition ball B(r ) = { x G d(x, e) r } G has polynomial growth if #B(r ) r d ( d N) (does not depend on choice of generating set) Example G = Z 2 : #B(r ) r 2. So Z 2 has polynomial growth. Easy: abelian grps have polynomial growth. Exer: Nilpotent grps have polynomial growth. Exer: Spse H is finite index subgroup of G. G has poly growth H has poly growth. Cor: Virtually nilpotent grps have poly growth. Virtually finite index subgrp is. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

4 Cor. Virtually nilp groups have polynomial growth. Example G = free group F 2. S = {a ±1, b ±1 }. #B(r ) is exp l r d. So F 2 does not have polynomial growth. b a 1 e a b 1 Exer: Groups of polynomial growth are amenable. Theorem (Gromov) Groups of polynomial growth are virtually nilpotent. Assume: G has polynomial growth. Prove: G is nilpotent * solvable. * (polycyclic) Exer. polycyclic with poly growth nilpotent *. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

5 Assume G has poly growth. Prove G is solvable. * Key Lemma f.d. rep ρ : G GL(V ), s.t. ρ(g) is infinite. Cor. Gromov s Theorem. Proof. #B ρ(g) (r ) #B G (r ) r d ρ(g) has polynomial growth no free subgrps ρ(g) is solvable * Tits alternative #B ker ρ (r ) #B G(2r ) #B ρ(g) (r ) r d r = r d 1. Induction on d: ker ρ is solvable *. ker ρ, ρ(g) solvable * G solvable *. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

6 Key Lem. f.d. rep ρ : G GL(V ), s.t. #ρ(g) =. Gromov s proof assumes hard theorems. (structure theory of locally cpct grps: Hilbert s 5th problem) B. Kleiner: much more elementary proof (ideas from Geometric Analysis: Analysis + Differential Geometry) Let V = { Lipschitz, harmonic f : G R }. f (x) f (y) C d(x, y), C R f (x) = 1 #S s S f (xs) G acts by translations: f g (x) = f (gx). Theorem (Kleiner, Colding Miniccozi) dim V < if G has polynomial growth. Prop. f V not constant. (do not need poly growth) So f G is infinite. (nonconstant harmonic func has no max) Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

7 Thm. Cor. dim { Lipschitz, harmonic f } <. Assume #B(r ) r d. #B(2r ) C #B(r ), r. bounded doubling (#B(r ) r d bdd dbling for most r.) Defn. For X G, f 2 2,X = x X f (x) 2. Theorem ϵ, k, C, r, set B of C balls of radius r : f harmonic with mean 0 on each B B f 2,B(8kr ) ϵk f 2,B(kr ). Idea of proof of corollary. f Lipschitz + bounded doubling f 2,B(r ) is poly f 2,B(8R) C f 2,B(R). Let R = kr. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

8 Analysis on G Thm. set B of C balls of radius r : f harmonic with mean 0 on each B B f 2,B(8kr ) ϵk f 2,B(kr ). Definition Let f : G R. s f (x) = f (xs) f (x). s f C: f is Lipschitz. f (x) f (y) C d(x, y). f (x) = #S f (x) s S f (xs). f = 0: f is harmonic. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

9 Thm. set B of C balls of radius r : f harmonic with mean 0 on each B B f 2,B(8kr ) ϵk f 2,B(kr ). Step 1 (Poincaré Inequality) Calculus: f (x) x 0 f (t) dt if f (0) = 0. f 2 2,B(r ) C 1 r 2 s S s f 2 2,B(3r ) Idea of proof. if mean 0 on B(r ) s d(x, x 0 ) = n: x 1 s 0 x 2 s3 1 x 2 sn x n = x. f (x) = f (x 0 ) + n i=1 s i f (x i ) n i=1 s i f (x i ). f (x) 2 ( n i=1 s i f (x i ) ) 2 n n i=1( si f (x i ) ) 2. Sum over x B(r ). (Note that n 2r.) Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

10 Step 2 (Reverse Poincaré Inequality) s S sf 2 2,B(R) C 2 R 2 f 2 2,B(3R) Proof 1 0 B(R) e B(2R) if f harmonic. ψ sψ 1 r Product rule (φψ) = φ ψ + s φ ψ ( s φ(x) = φ(xs)) (f ψ 2 ) = f ψ 2 + s f (ψ ψ + s ψ ψ ) = f ψ 2 + s f ψ ( ψ + 2ψ ) φ ψ X = x X φ(x) ψ(x) f (f ψ 2 ) = f ψ 2 2,G + C R 2 f s f B(2R) + C R f s f ψ. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

11 f (f ψ 2 ) = f ψ 2 2,G + C R 2 f s f B(2R) + C R f s f ψ. Reverse P I: s S s f 2 2,B(R) C R 2 f 2 2,B(3R) Sum over s S: 0 = ( LHS) + ( RHS) + (smaller). s s f s (f ψ 2 ) = s s 1 s f f ψ 2 = 2 f f ψ 2 = 2 0 f ψ 2 = 0. Schwarz Inequality: φ ψ φ 2 ψ 2 f s f B(2R) = s f f s f B(2R) s f s f B(2R) + f s f B(2R) f 2 2,B(2R+1) + f 2,B(2R) s f 2,B(2R) = 2 f 2 2,B(2R+1). Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

12 f (f ψ 2 ) = f ψ 2 2,G + C R 2 f s f B(2R) + C R f s f ψ. Reverse P I: s S s f 2 2,B(R) C 2 R 2 f 2 2,B(3R) Sum over s S: 0 = ( LHS) + ( RHS) + (smaller) C R f s f ψ = f ψ C s f R B(2R) s f 2,B(2R) f ψ 2 C R 1 2 f ψ C R 2 s f 2 2,B(2R) 1 2 f ψ C R 2 f 2 2,B(2R+1). (Schwarz Inequality) Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

13 1. Poincaré : f 2 2,B(r ) C 1 r 2 s S s f 2 2,B(3r ). 2. Reverse P I: s S s f 2 2,B(R) C 2 R 2 f 2 2,B(3R). Step 3 (finish): f 2,B(8kr ) ϵk f 2,B(kr ) Vitali Covering Lemma: ϵ, k, C, r, B Cover B(kr ) with #B(2kr ) #B(r /2) < C balls of radius r. No pt in more than #B(4r ) < C tripled balls. #B(r /2) f 2 2,B(kr ) B B f 2 2,B (balls cover B(kr )) C 1 r 2 s S B B s f 2 2,3B CC 1 r 2 s S s f 2 2,B(2kr ) CC 1 r 2 C 2 (2kr ) 2 f 2 2,B(3(2kr )) (Poincaré Ineq) ( ) no pt in more than C balls ( ) Reverse Poincaré Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

14 Review of Kleiner s proof Assume G has polynomial growth. Poincaré. f 2 2,B(r ) C r 2 s S s f 2 2,B(3r ). Reverse P I. s S sf 2 2,B(R) C R 2 f 2 2,B(3R). Thm. f 2,B(8kr ) ϵk f 2,B(kr ). Cor. dim { Lipschitz, harmonic f } <. Prop. Lipschitz, harmonic, nonconstant f. Key Lem. f.d. rep ρ : G GL(V ), s.t. #ρ(g) =. Gromov s Theorem. G is virtually nilpotent. Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15

15 Expository D. W. Morris: Groups of polynomial growth (after Gromov, Kleiner, and Shalom Tao). (on arxiv soon) T. Tao, A proof of Gromov s theorem. a proof of gromovs theorem/ Research B. Kleiner: A new proof of Gromov s theorem on groups of polynomial growth, J. Amer. Math. Soc. 23 (2010) MR , Y. Shalom and T. Tao: A finitary version of Gromov s polynomial growth theorem, Geom. Funct. Anal. 20 (2010) MR , Dave Witte Morris (Univ. of Lethbridge) Groups of polynomial growth U of Virginia, April / 15