Growth of quotients of Kleinian groups

Transcription

1 rowth of quotients of Kleinian groups Andrea Sambusetti - University La Sapienza of Rome, Italy joint work with F. Dal bo - University of Rennes, France M. Peigné - University of Tours, France J.C. Picaud - University of Tours, France Spectral Theory and eometry Institut Fourier, renoble 1-5 June, 2009

2 Outline 1 2

3 rowth tightness for word metrics Context: discrete group endowed with a generating set S is growth tight if δ Γ < δ for every proper quotient Γ of Asymptotic characterization of free groups (folklore) Let = F n /N be a group on n generators S = {g 1,..., g n }. Then δ (,S) δ Fn = log(2n 1) and δ = log(2n 1) is free on S. S.,02 Free and amalgamated products (over finite subgroups) are growth tight. Application: first examples of groups of exponential growth whose minimal algebraic entropy Entalg() = inf S δ (,S) is not attained by any generating set S. Arz.-Lys., 02 romov hyperbolic groups are growth tight (w.r. to word metric).

4 rowth tightness in Riemannian geometry [S.,08] Cocompact Kleinian groups are growth tight If X and = /N X = N\X, then the gap Ω(X) = Ent(X) Ent(X) = δ δ can be estimated in terms of X 0 = \X and sys(n). Applications to systoles of open coverings of compact manifolds: [S.,08] Let X X 0 be a normal covering of X 0 with k(x 0 ) 1: [ ] 1 sys(x) C(n, r 0, v 0 ) log Ω(X) where inj(x 0 ) r 0, vol(x 0 ) v 0 [S.,02] Let S S be a normal covering of a hyperbolic surface S: [ ] sys(s) 1 12 log 2(g 1) 2 D Ω(S) where g = genus(s), D = diam(s).

5 Applications to the bottom of the spectrum: [S.,08] Let X X 0 be a normal covering of a compact, locally symmetric manifold X 0 on K with negative curvature 4 k(x 0 ) 1. λ 0 (X) λ 0 (KH m ) 1 4 e 2c K,m(v 0 ) sys(x) ( ) Ent(X) 1 Ent(KH m ) + e c K,m(v 0 ) sys(x) 2 where vol(x 0 ) v 0 and c K,m (v 0 ) is an explicit function. Open problems The spectrum of critical exponents Let be a cocompact/convex-cocompact/geometrically finite Kleinian group The AP Is δ isolated in Exp()? Does there exist with Ω() = inf{δ δ proper quotient of } > 0? Let be a geometrically finite Kleinian group of X satisfying (PC): estimate Ω = δ δ in terms of sys hyp (N) and of the Nielsen core of \X

6 Kleinian group with Vol(\X) < = N\ quotient acting on X = N\X

7 Kleinian group with Vol(\X) < = N\ quotient acting on X = N\X

8 Kleinian group with Vol(\X) < = N\ quotient acting on X = N\X x, D, K, C i and P project to x, D, K, C i and P

9 Kleinian group with Vol(\X) < = N\ quotient acting on X = N\X x, D, K, C i and P project to x, D, K, C i and P A σ ( x, R) = {g R σ < d( x, g x) R + σ} v σ ( x, R) = card Aσ ( x, R)

10 Kleinian group with Vol(\X) < = N\ quotient acting on X = N\X x, D, K, C i and P project to x, D, K, C i and P A σ ( x, R) = {g R σ < d( x, g x) R + σ} v σ ( x, R) = card Aσ ( x, R) These series are equivalent: P ( x, δ ) = g e δ d( x,g x) P ( x, δ ) = e δ n v σ ( x, n) n 0

11 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D

12 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): ( x, N + M) ( x, N) ( x, M) (1)

13 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): which will imply: ( x, N + M) ( x, N) ( x, M) (1) lim R 1 R ln v 2D ( x, R) = δ ( x, R) e δ R for R 0

14 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): which will imply: ( x, N + M) ( x, N) ( x, M) (1) 1 lim R R ln v 2D ( x, R) = δ ( x, R) e δ R for R 0 the series P ( x, δ ) = n 0 e δ n v σ ( x, n) is divergent

15 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): Why? ( x, N + M) ( x, N) ( x, M) (1)

16 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): ( x, N + M) ( x, N) ( x, M) (1) g x A 2D ( x, N + M)

17 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): ( x, N + M) ( x, N) ( x, M) (1) g x A 2D ( x, N + M)

18 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): ( x, N + M) ( x, N) ( x, M) (1) g x A 2D ( x, N + M)

19 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): ( x, N + M) ( x, N) ( x, M) (1) g x A 2D ( x, N + M) h A 2D ( x, N) and g A 2D (h x, M)

20 Sketch of proof when is cocompact We have X 0 = \X = \X compact with diameter D We prove (argument of.robert): ( x, N + M) ( x, N) ( x, M) (1) g x A 2D ( x, N + M) h A 2D ( x, N) and g A 2D (h x, M) A 2D ( x, N+M) which proves (1). 2D A h A 2D( x,n) (h x, M)

21 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ δ < δ for all parabolic P < We would like to prove: ( x, N + M) C v 2D( x, N) v 2D( x, M) e δn e δm (2)

22 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ δ < δ for all parabolic P < We actually prove: ( ( x, N + M) N C n=1 ) ( ( x, n) M e δn m=1 ) ( x, m) e δm

23 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ δ < δ for all parabolic P < We prove: ( x, N + M) } {{} N C n=1 ( x, n) M e δn m=1 }{{}}{{} w N+M W N W M ( x, m) (2) e δm Setting w n = ( x,n+m) and W n = n k=1 w k = w N+M C W N W M

24 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ δ < δ for all parabolic P < We prove: ( x, N + M) } {{} N C n=1 ( x, n) M e δn m=1 }{{}}{{} w N+M W N W M ( x, m) (2) e δm Setting w n = ( x,n+m) and W n = n k=1 w k = w N+M C W N W M This implies: lim n 1 n ln(w n) = lim n 1 n ln(w n) = δ δ = ɛ > 0 W n c eɛn n for n 0

25 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ δ < δ for all parabolic P < We prove: ( x, N + M) } {{} N C n=1 ( x, n) M e δn m=1 }{{}}{{} w N+M W N W M ( x, m) (2) e δm Setting w n = ( x,n+m) and W n = n k=1 w k = w N+M C W N W M This implies: lim n 1 n ln(w n) = lim n 1 n ln(w n) = δ δ = ɛ > 0 W n c eɛn n for n 0 P ( x, δ ) = n 0 v σ ( x,n) e δ n = n 0 v σ ( x,n)e ɛn e (δ ɛ)n

26 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ δ < δ for all parabolic P < We prove: ( x, N + M) } {{} N C n=1 ( x, n) M e δn m=1 }{{}}{{} w N+M W N W M ( x, m) (2) e δm Setting w n = ( x,n+m) and W n = n k=1 w k = w N+M C W N W M This implies: lim n 1 n ln(w n) = lim n 1 n ln(w n) = δ δ = ɛ > 0 W n c eɛn n for n 0 P ( x, δ ) = n 1 v σ ( x,n) e δ n = n 1 w ne ɛn n 1 W ne ɛn divergent.

27 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2)

28 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M)

29 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X

30 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K

31 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1

32 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1 h 1 A 2D (x, n) and g A 2D (h 2 x, m)

33 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1 h 1 A 2D (x, n) and g A 2D (h 2 x, m) A 2D ( x, N + M) n=1,...,n p B (N+M n m+3d) m=1,...,m Pi i=1,...,l h A 2D ( x,n) A 2D (hp x, m)

34 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1 h 1 A 2D (x, n) and g A 2D (h 2 x, m) A 2D ( x, N + M) n=1,...,n ( x, N + M) N M n=1 m=1 i=1 p B (N+M n m+3d) m=1,...,m Pi i=1,...,l l h A 2D ( x,n) A 2D (hp x, m) 2D ( x, n)v ( x, m)v ( x, N n + M m + 3D) P i

35 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C A 2D ( x, N + M) n=1,...,n ( x, N + M) N M δ < δ for all parabolic P < N ( x,n) n=1 e δn p B (N+M n m+3d) m=1,...,m Pi i=1,...,l l n=1 m=1 i=1 M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1 h 1 A 2D (x, n) and g A 2D (h 2 x, m) and as v Pi (R) ce δr h A 2D ( x,n) A 2D (hp x, m) 2D ( x, n)v ( x, m)v ( x, N n + M m + 3D) P i

36 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C A 2D ( x, N + M) n=1,...,n ( x, N + M) C N δ < δ for all parabolic P < N ( x,n) n=1 e δn p B (N+M n m+3d) m=1,...,m Pi i=1,...,l M n=1 m=1 M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1 h 1 A 2D (x, n) and g A 2D (h 2 x, m) and as v Pi (R) ce δr h A 2D ( x,n) ( x, n)eδ(n n) A 2D (hp x, m) ( x, m)eδ(m m)

37 Proof for not cocompact, but Vol(\X) < ɛ satisfies (PC): δ P < {}}{ Let us show: v2d ( x,n+m) C δ < δ for all parabolic P < N ( x,n) n=1 e δn M ( x,m) m=1 e δm (2) start with g x A 2D ( x, N + M) lift everything to X y 1 h 1 K and y 2 h 2 K h 2 = h 1 p for some p hp i h 1 h 1 A 2D (x, n) and g A 2D (h 2 x, m) and as v Pi (R) ce δr ( x, N + M) C N M n=1 m=1 ( x, n)eδ(n n) ( x, m)eδ(m m) (2)