geodesic flows on quotients of bruhat-tits buildings - and ... · geodesic flows on quotients of...
TRANSCRIPT
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Geodesic flows on quotients of Bruhat-Titsbuildings
and Number Theory
Anish Ghosh
University of East Anglia
joint with J. Athreya (Urbana) and A. Prasad (Chennai)
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Diagonal Actions
Recurrence
Buildings
Shrinking Targets
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Diagonal Actions
Recurrence
Buildings
Shrinking Targets
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Diagonal Actions
Recurrence
Buildings
Shrinking Targets
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Diagonal Actions
Recurrence
Buildings
Shrinking Targets
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Shrinking
Target
Problems
0− 1 laws
Return time asymptotics
Heirarchy
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Shrinking
Target
Problems
0− 1 laws
Return time asymptotics
Heirarchy
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Shrinking
Target
Problems
0− 1 laws
Return time asymptotics
Heirarchy
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Shrinking
Target
Problems
0− 1 laws
Return time asymptotics
Heirarchy
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Shrinking
Target
Problems
0− 1 laws
Return time asymptotics
Heirarchy
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Shrinking
Target
Problems
0− 1 laws
Return time asymptotics
Heirarchy
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Excursions in hyperbolic 3 manifolds
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Sullivan (1982)
Y = Hk+1/Γ, y0 ∈ Y , {rt} a sequence of real numbers
For any y ∈ Y and almost every ξ ∈ Ty (Y ) there are infinitely manyt ∈ N such that
dist(y0, γt(y , ξ)) ≥ rt
if and only if
∞∑t=1
e−krt =∞
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Sullivan (1982)
Y = Hk+1/Γ, y0 ∈ Y , {rt} a sequence of real numbers
For any y ∈ Y and almost every ξ ∈ Ty (Y ) there are infinitely manyt ∈ N such that
dist(y0, γt(y , ξ)) ≥ rt
if and only if
∞∑t=1
e−krt =∞
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Sullivan (1982)
Y = Hk+1/Γ, y0 ∈ Y , {rt} a sequence of real numbers
For any y ∈ Y and almost every ξ ∈ Ty (Y ) there are infinitely manyt ∈ N such that
dist(y0, γt(y , ξ)) ≥ rt
if and only if
∞∑t=1
e−krt =∞
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Logarithm Laws
Corollary
For almost every y ∈ Y and almost all ξ ∈ Ty (Y )
lim supt→∞
dist(y , γt(y , ξ))
log t=
1
k
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Some Number theory
For almost every z , there exist infinitely many pairs (p, q) ∈ O(√−d)
such that
|z − p
q| < ψ(|q|)
|q|2and (p, q) = O(
√−d)
if and only if∞∑t=1
ψ(t)
t2=∞
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Some Number theory
For almost every z , there exist infinitely many pairs (p, q) ∈ O(√−d)
such that
|z − p
q| < ψ(|q|)
|q|2and (p, q) = O(
√−d)
if and only if∞∑t=1
ψ(t)
t2=∞
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition ofMargulis)
Along with geometric arguments
To force quasi-independence
And then apply the Borel-Cantelli Lemma
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition ofMargulis)
Along with geometric arguments
To force quasi-independence
And then apply the Borel-Cantelli Lemma
20 / 127
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition ofMargulis)
Along with geometric arguments
To force quasi-independence
And then apply the Borel-Cantelli Lemma
21 / 127
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition ofMargulis)
Along with geometric arguments
To force quasi-independence
And then apply the Borel-Cantelli Lemma
22 / 127
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γnon-uniform lattice
Y = K\G/Γ a locally symmetric space
Logarithm laws for locally symmetric spaces
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γnon-uniform lattice
Y = K\G/Γ a locally symmetric space
Logarithm laws for locally symmetric spaces
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γnon-uniform lattice
Y = K\G/Γ a locally symmetric space
Logarithm laws for locally symmetric spaces
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γnon-uniform lattice
Y = K\G/Γ a locally symmetric space
Logarithm laws for locally symmetric spaces
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Khintchine-Groshev Theorem
For almost all A ∈ Matm×n(R), there exist infinitely manyq ∈ Zn,p ∈ Zm such that
‖Aq + p‖m < ψ(‖q‖n)
if and only if
∞∑t=1
ψ(t) =∞
If time permits
More general multiplicative results
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Khintchine-Groshev Theorem
For almost all A ∈ Matm×n(R), there exist infinitely manyq ∈ Zn,p ∈ Zm such that
‖Aq + p‖m < ψ(‖q‖n)
if and only if
∞∑t=1
ψ(t) =∞
If time permits
More general multiplicative results
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Khintchine-Groshev Theorem
For almost all A ∈ Matm×n(R), there exist infinitely manyq ∈ Zn,p ∈ Zm such that
‖Aq + p‖m < ψ(‖q‖n)
if and only if
∞∑t=1
ψ(t) =∞
If time permits
More general multiplicative results
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SLn(R)/ SLn(Z) can be identified with the space Xn of unimodularlattices in Rn
Mahler’s compactness criterion describes compact subsets of Xn
The systole of a lattice
s(Λ) := minv∈Λ‖v‖
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SLn(R)/ SLn(Z) can be identified with the space Xn of unimodularlattices in Rn
Mahler’s compactness criterion describes compact subsets of Xn
The systole of a lattice
s(Λ) := minv∈Λ‖v‖
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SLn(R)/ SLn(Z) can be identified with the space Xn of unimodularlattices in Rn
Mahler’s compactness criterion describes compact subsets of Xn
The systole of a lattice
s(Λ) := minv∈Λ‖v‖
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Dani Correspondence
x ! ux :=
(1 x0 1
)
gt :=
(et 00 e−t
)
Read Diophantine properties of x from the gt (semi) orbits of uxZ2
For example, x is badly approximable if and only if the orbit of uxZ2
is bounded
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Dani Correspondence
x ! ux :=
(1 x0 1
)
gt :=
(et 00 e−t
)
Read Diophantine properties of x from the gt (semi) orbits of uxZ2
For example, x is badly approximable if and only if the orbit of uxZ2
is bounded
34 / 127
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Dani Correspondence
x ! ux :=
(1 x0 1
)
gt :=
(et 00 e−t
)
Read Diophantine properties of x from the gt (semi) orbits of uxZ2
For example, x is badly approximable if and only if the orbit of uxZ2
is bounded
35 / 127
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Dani Correspondence
x ! ux :=
(1 x0 1
)
gt :=
(et 00 e−t
)
Read Diophantine properties of x from the gt (semi) orbits of uxZ2
For example, x is badly approximable if and only if the orbit of uxZ2
is bounded
36 / 127
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More generally
There are infinitely many solutions to
|qx + p| < ψ(q)
if and only if there are infinitely many t > 0 such that
s(gtuxZ2) ≤ r(t)
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More generally
There are infinitely many solutions to
|qx + p| < ψ(q)
if and only if there are infinitely many t > 0 such that
s(gtuxZ2) ≤ r(t)
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r(t) defines the complement Xn(t) of a compact set
Shrinking systolic neighborhoods
If r(t)→ 0 very fast we should expect few solutions
The speed is governed by convergence of
∞∑t=0
vol(Xn(t))
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r(t) defines the complement Xn(t) of a compact set
Shrinking systolic neighborhoods
If r(t)→ 0 very fast we should expect few solutions
The speed is governed by convergence of
∞∑t=0
vol(Xn(t))
40 / 127
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r(t) defines the complement Xn(t) of a compact set
Shrinking systolic neighborhoods
If r(t)→ 0 very fast we should expect few solutions
The speed is governed by convergence of
∞∑t=0
vol(Xn(t))
41 / 127
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r(t) defines the complement Xn(t) of a compact set
Shrinking systolic neighborhoods
If r(t)→ 0 very fast we should expect few solutions
The speed is governed by convergence of
∞∑t=0
vol(Xn(t))
42 / 127
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Similar story for logarithm laws
Model geodesic flow on T 1(K\G/Γ) by the action of diagonalone-parameter subgroups on G/Γ
Compute vol({x ∈ G/Γ : d(x0, gtx) ≥ r(t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients forautomorphic representation
For φ, ψ ∈ L2(G/Γ), and g ∈ G
|〈gφ, ψ〉| � L2l (φ)L2
l (ψ)H(g)−λ
To force quasi-independence on average
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Similar story for logarithm laws
Model geodesic flow on T 1(K\G/Γ) by the action of diagonalone-parameter subgroups on G/Γ
Compute vol({x ∈ G/Γ : d(x0, gtx) ≥ r(t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients forautomorphic representation
For φ, ψ ∈ L2(G/Γ), and g ∈ G
|〈gφ, ψ〉| � L2l (φ)L2
l (ψ)H(g)−λ
To force quasi-independence on average
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Similar story for logarithm laws
Model geodesic flow on T 1(K\G/Γ) by the action of diagonalone-parameter subgroups on G/Γ
Compute vol({x ∈ G/Γ : d(x0, gtx) ≥ r(t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients forautomorphic representation
For φ, ψ ∈ L2(G/Γ), and g ∈ G
|〈gφ, ψ〉| � L2l (φ)L2
l (ψ)H(g)−λ
To force quasi-independence on average
45 / 127
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Similar story for logarithm laws
Model geodesic flow on T 1(K\G/Γ) by the action of diagonalone-parameter subgroups on G/Γ
Compute vol({x ∈ G/Γ : d(x0, gtx) ≥ r(t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients forautomorphic representation
For φ, ψ ∈ L2(G/Γ), and g ∈ G
|〈gφ, ψ〉| � L2l (φ)L2
l (ψ)H(g)−λ
To force quasi-independence on average
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Similar story for logarithm laws
Model geodesic flow on T 1(K\G/Γ) by the action of diagonalone-parameter subgroups on G/Γ
Compute vol({x ∈ G/Γ : d(x0, gtx) ≥ r(t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients forautomorphic representation
For φ, ψ ∈ L2(G/Γ), and g ∈ G
|〈gφ, ψ〉| � L2l (φ)L2
l (ψ)H(g)−λ
To force quasi-independence on average
47 / 127
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Similar story for logarithm laws
Model geodesic flow on T 1(K\G/Γ) by the action of diagonalone-parameter subgroups on G/Γ
Compute vol({x ∈ G/Γ : d(x0, gtx) ≥ r(t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients forautomorphic representation
For φ, ψ ∈ L2(G/Γ), and g ∈ G
|〈gφ, ψ〉| � L2l (φ)L2
l (ψ)H(g)−λ
To force quasi-independence on average
48 / 127
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Much recent activity
Maucourant, Hersonsky-Paulin, Parkonnen-Paulin, Chaika
Nogueira, Laurent-Nogueira, G-Gorodnik-Nevo
49 / 127
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Much recent activity
Maucourant, Hersonsky-Paulin, Parkonnen-Paulin, Chaika
Nogueira, Laurent-Nogueira, G-Gorodnik-Nevo
50 / 127
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Much recent activity
Maucourant, Hersonsky-Paulin, Parkonnen-Paulin, Chaika
Nogueira, Laurent-Nogueira, G-Gorodnik-Nevo
51 / 127
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Key technical point
In order to apply decay of matrix coefficients we have to smoothencusp neighborhoods
And use suitable Sobolev norms
This destroys sensitive information about exponents
Which is crucial in G-Gorodnik-Nevo
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Key technical point
In order to apply decay of matrix coefficients we have to smoothencusp neighborhoods
And use suitable Sobolev norms
This destroys sensitive information about exponents
Which is crucial in G-Gorodnik-Nevo
53 / 127
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Key technical point
In order to apply decay of matrix coefficients we have to smoothencusp neighborhoods
And use suitable Sobolev norms
This destroys sensitive information about exponents
Which is crucial in G-Gorodnik-Nevo
54 / 127
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Key technical point
In order to apply decay of matrix coefficients we have to smoothencusp neighborhoods
And use suitable Sobolev norms
This destroys sensitive information about exponents
Which is crucial in G-Gorodnik-Nevo
55 / 127
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
dist metric on G , ¯dist metric on G/Γ defined by
¯dist(gΓ, hΓ) := infγ∈Γ{dist(gγ, h) : γ ∈ Γ}
Coarse isometry (Leuzinger + Ji)
¯dist(gΓ, hΓ) ≤ dist(g , h) ≤ ¯dist(gΓ, hΓ) + C
For g , h ∈ S
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
dist metric on G , ¯dist metric on G/Γ defined by
¯dist(gΓ, hΓ) := infγ∈Γ{dist(gγ, h) : γ ∈ Γ}
Coarse isometry (Leuzinger + Ji)
¯dist(gΓ, hΓ) ≤ dist(g , h) ≤ ¯dist(gΓ, hΓ) + C
For g , h ∈ S
57 / 127
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
dist metric on G , ¯dist metric on G/Γ defined by
¯dist(gΓ, hΓ) := infγ∈Γ{dist(gγ, h) : γ ∈ Γ}
Coarse isometry (Leuzinger + Ji)
¯dist(gΓ, hΓ) ≤ dist(g , h) ≤ ¯dist(gΓ, hΓ) + C
For g , h ∈ S
58 / 127
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
dist metric on G , ¯dist metric on G/Γ defined by
¯dist(gΓ, hΓ) := infγ∈Γ{dist(gγ, h) : γ ∈ Γ}
Coarse isometry (Leuzinger + Ji)
¯dist(gΓ, hΓ) ≤ dist(g , h) ≤ ¯dist(gΓ, hΓ) + C
For g , h ∈ S
59 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
60 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
61 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
62 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
63 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
64 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
65 / 127
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Trees and Buildings
If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
Example from Shahar Mozes’ talk
G = PGL2(Qp)× PGL2(Qq)
Γ a co-compact lattice coming from a quaternion algebra
X/Γ is a finite graph
In fact, any lattice in a p-adic Lie group is necessarily cocompact(Tamagawa)
66 / 127
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Trees and Buildings
However, we could consider G = PGL2(Fp((t)))× PGL2(Fp((t−1)))
And Γ = PGL2(Fp[t, t−1])
Which turns out to be a non-uniform lattice in G
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Trees and Buildings
However, we could consider G = PGL2(Fp((t)))× PGL2(Fp((t−1)))
And Γ = PGL2(Fp[t, t−1])
Which turns out to be a non-uniform lattice in G
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Trees and Buildings
However, we could consider G = PGL2(Fp((t)))× PGL2(Fp((t−1)))
And Γ = PGL2(Fp[t, t−1])
Which turns out to be a non-uniform lattice in G
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Trees and Buildings
The Tree of SL2(F2((X−1)))
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Trees and Buildings
Ultrametric logarithm laws
So it makes sense to investigate analogues of logarithm laws for
The graph K\G/Γ where G is an algebraic group
Defined over a totally disconnected field
K is a compact open subgroup and Γ is a lattice in G
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Trees and Buildings
Ultrametric logarithm laws
So it makes sense to investigate analogues of logarithm laws for
The graph K\G/Γ where G is an algebraic group
Defined over a totally disconnected field
K is a compact open subgroup and Γ is a lattice in G
72 / 127
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Trees and Buildings
Ultrametric logarithm laws
So it makes sense to investigate analogues of logarithm laws for
The graph K\G/Γ where G is an algebraic group
Defined over a totally disconnected field
K is a compact open subgroup and Γ is a lattice in G
73 / 127
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Trees and Buildings
Ultrametric logarithm laws
So it makes sense to investigate analogues of logarithm laws for
The graph K\G/Γ where G is an algebraic group
Defined over a totally disconnected field
K is a compact open subgroup and Γ is a lattice in G
74 / 127
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Trees and Buildings
Local fields of positive characteristic
Fp - finite field
Fp[X ] ! Z
Fp(X ) ! Q
Fp((X−1)) ! R
75 / 127
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Trees and Buildings
Local fields of positive characteristic
Fp - finite field
Fp[X ] ! Z
Fp(X ) ! Q
Fp((X−1)) ! R
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Trees and Buildings
Local fields of positive characteristic
Fp - finite field
Fp[X ] ! Z
Fp(X ) ! Q
Fp((X−1)) ! R
77 / 127
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Trees and Buildings
Local fields of positive characteristic
Fp - finite field
Fp[X ] ! Z
Fp(X ) ! Q
Fp((X−1)) ! R
78 / 127
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Trees and Buildings
Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
With a family of subcomplexes called apartments
There is a nice (CAT(0)) metric on the building
And G acts by isometries
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Trees and Buildings
Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
With a family of subcomplexes called apartments
There is a nice (CAT(0)) metric on the building
And G acts by isometries
80 / 127
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Trees and Buildings
Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
With a family of subcomplexes called apartments
There is a nice (CAT(0)) metric on the building
And G acts by isometries
81 / 127
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Trees and Buildings
Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
With a family of subcomplexes called apartments
There is a nice (CAT(0)) metric on the building
And G acts by isometries
82 / 127
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Trees and Buildings
Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
With a family of subcomplexes called apartments
There is a nice (CAT(0)) metric on the building
And G acts by isometries
83 / 127
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Trees and Buildings
Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
With a family of subcomplexes called apartments
There is a nice (CAT(0)) metric on the building
And G acts by isometries
84 / 127
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Trees and Buildings
Borel-Harish-Chandra, Behr-Harder
G (Fp[X ]) is a lattice in G (Fp((X−1)))
Lubotzky
G (rank 1) has non-uniform and uniform, arithmetic and non-arithmeticlattices.
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Trees and Buildings
Borel-Harish-Chandra, Behr-Harder
G (Fp[X ]) is a lattice in G (Fp((X−1)))
Lubotzky
G (rank 1) has non-uniform and uniform, arithmetic and non-arithmeticlattices.
86 / 127
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
88 / 127
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
89 / 127
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
90 / 127
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
91 / 127
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
92 / 127
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Trees and Buildings
Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
Hersonsky and Paulin obtained logarithm laws for T/Γ
Where Γ is a geometrically finite lattice in Aut(T )
By Serre + Raghunathan + Lubotzky
This includes all algebraic examples
Namely lattices in rank 1 algebraic groups
93 / 127
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Trees and Buildings
Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
Their method is very geometric
And reminiscent of Sullivan
94 / 127
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Trees and Buildings
Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
Their method is very geometric
And reminiscent of Sullivan
95 / 127
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Trees and Buildings
Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
Their method is very geometric
And reminiscent of Sullivan
96 / 127
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Trees and Buildings
Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
Their method is very geometric
And reminiscent of Sullivan
97 / 127
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Trees and Buildings
Let G be a connected, semisimple linear algebraic group defined andsplit over Fp
Let Γ be an S arithmetic lattice in GS
By a result of Margulis + Venkataramana
This includes any irreducible lattice in higher rank
98 / 127
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Trees and Buildings
Let G be a connected, semisimple linear algebraic group defined andsplit over Fp
Let Γ be an S arithmetic lattice in GS
By a result of Margulis + Venkataramana
This includes any irreducible lattice in higher rank
99 / 127
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Trees and Buildings
Let G be a connected, semisimple linear algebraic group defined andsplit over Fp
Let Γ be an S arithmetic lattice in GS
By a result of Margulis + Venkataramana
This includes any irreducible lattice in higher rank
100 / 127
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Trees and Buildings
Let G be a connected, semisimple linear algebraic group defined andsplit over Fp
Let Γ be an S arithmetic lattice in GS
By a result of Margulis + Venkataramana
This includes any irreducible lattice in higher rank
101 / 127
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Trees and Buildings
And their quotients
We prove analogues of the results of Kleinbock-Margulis
Namely logarithm laws for arithmetic quotients of
Bruhat-Tits buildings of semisimple groups
And several results in Diophantine Approximation
102 / 127
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Trees and Buildings
And their quotients
We prove analogues of the results of Kleinbock-Margulis
Namely logarithm laws for arithmetic quotients of
Bruhat-Tits buildings of semisimple groups
And several results in Diophantine Approximation
103 / 127
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Trees and Buildings
And their quotients
We prove analogues of the results of Kleinbock-Margulis
Namely logarithm laws for arithmetic quotients of
Bruhat-Tits buildings of semisimple groups
And several results in Diophantine Approximation
104 / 127
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Trees and Buildings
And their quotients
We prove analogues of the results of Kleinbock-Margulis
Namely logarithm laws for arithmetic quotients of
Bruhat-Tits buildings of semisimple groups
And several results in Diophantine Approximation
105 / 127
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Trees and Buildings
Main Result
Let B be a family of measurable subsets and F = {fn} denote asequence of µ-preserving transformations of G/Γ
B is Borel-Cantelli for F if for every sequence {An : n ∈ N} of setsfrom B
µ({x ∈ G/Γ | fn(x) ∈ An for infinitely many n ∈ N})
=
0 if
∑∞n=1 µ(An) <∞
1 if∑∞
n=1 µ(An) =∞
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Trees and Buildings
Main Result
Let B be a family of measurable subsets and F = {fn} denote asequence of µ-preserving transformations of G/Γ
B is Borel-Cantelli for F if for every sequence {An : n ∈ N} of setsfrom B
µ({x ∈ G/Γ | fn(x) ∈ An for infinitely many n ∈ N})
=
0 if
∑∞n=1 µ(An) <∞
1 if∑∞
n=1 µ(An) =∞
107 / 127
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Trees and Buildings
∆ a function on G/Γ define the tail distribution function
Φ∆(n) := µ({x ∈ G/Γ | ∆(x) ≥ sn})
Call ∆ “distance-like” if
Φ∆(n) � s−κn ∀ n ∈ Z
108 / 127
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Trees and Buildings
∆ a function on G/Γ define the tail distribution function
Φ∆(n) := µ({x ∈ G/Γ | ∆(x) ≥ sn})
Call ∆ “distance-like” if
Φ∆(n) � s−κn ∀ n ∈ Z
109 / 127
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Trees and Buildings
Let F = {fn | n ∈ N} be a sequence of elements of G satisfying
supm∈N
∞∑n=1
‖fnf −1m ‖−β <∞ ∀ β > 0,
and let ∆ be a “distance-like” function on G/Γ. Then
B(∆) := {{x ∈ G/Γ | ∆(x) ≥ sn} | n ∈ Z}
is Borel-Cantelli for F
110 / 127
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Trees and Buildings
Let F = {fn | n ∈ N} be a sequence of elements of G satisfying
supm∈N
∞∑n=1
‖fnf −1m ‖−β <∞ ∀ β > 0,
and let ∆ be a “distance-like” function on G/Γ. Then
B(∆) := {{x ∈ G/Γ | ∆(x) ≥ sn} | n ∈ Z}
is Borel-Cantelli for F
111 / 127
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Trees and Buildings
Let F = {fn | n ∈ N} be a sequence of elements of G satisfying
supm∈N
∞∑n=1
‖fnf −1m ‖−β <∞ ∀ β > 0,
and let ∆ be a “distance-like” function on G/Γ. Then
B(∆) := {{x ∈ G/Γ | ∆(x) ≥ sn} | n ∈ Z}
is Borel-Cantelli for F
112 / 127
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Trees and Buildings
Ergodic ideas
We establish effective decay of matrix coefficients for semisimplegroups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regularrepresentation of G on G/Γ
For simple groups this is known due to work of Bekka and Lubotzky
They also show that there are tree lattices without spectral gap
113 / 127
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Trees and Buildings
Ergodic ideas
We establish effective decay of matrix coefficients for semisimplegroups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regularrepresentation of G on G/Γ
For simple groups this is known due to work of Bekka and Lubotzky
They also show that there are tree lattices without spectral gap
114 / 127
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Trees and Buildings
Ergodic ideas
We establish effective decay of matrix coefficients for semisimplegroups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regularrepresentation of G on G/Γ
For simple groups this is known due to work of Bekka and Lubotzky
They also show that there are tree lattices without spectral gap
115 / 127
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Trees and Buildings
Ergodic ideas
We establish effective decay of matrix coefficients for semisimplegroups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regularrepresentation of G on G/Γ
For simple groups this is known due to work of Bekka and Lubotzky
They also show that there are tree lattices without spectral gap
116 / 127
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Trees and Buildings
Ergodic ideas
We establish effective decay of matrix coefficients for semisimplegroups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regularrepresentation of G on G/Γ
For simple groups this is known due to work of Bekka and Lubotzky
They also show that there are tree lattices without spectral gap
117 / 127
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Trees and Buildings
Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined andquasi-split over k and Γ be an non-uniform irreducible lattice in G . Forany g ∈ G and any smooth functions φ, ψ ∈ L2
0(G/Γ),
|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.
We use the Burger-Sarnak method as extended to p-adic and positivecharacteristic fields by
Clozel and Ullmo
Along with the solution of the Ramanujan conjecture for GL2 due toDrinfield
118 / 127
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Trees and Buildings
Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined andquasi-split over k and Γ be an non-uniform irreducible lattice in G . Forany g ∈ G and any smooth functions φ, ψ ∈ L2
0(G/Γ),
|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.
We use the Burger-Sarnak method as extended to p-adic and positivecharacteristic fields by
Clozel and Ullmo
Along with the solution of the Ramanujan conjecture for GL2 due toDrinfield
119 / 127
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Trees and Buildings
Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined andquasi-split over k and Γ be an non-uniform irreducible lattice in G . Forany g ∈ G and any smooth functions φ, ψ ∈ L2
0(G/Γ),
|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.
We use the Burger-Sarnak method as extended to p-adic and positivecharacteristic fields by
Clozel and Ullmo
Along with the solution of the Ramanujan conjecture for GL2 due toDrinfield
120 / 127
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Trees and Buildings
Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined andquasi-split over k and Γ be an non-uniform irreducible lattice in G . Forany g ∈ G and any smooth functions φ, ψ ∈ L2
0(G/Γ),
|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.
We use the Burger-Sarnak method as extended to p-adic and positivecharacteristic fields by
Clozel and Ullmo
Along with the solution of the Ramanujan conjecture for GL2 due toDrinfield
121 / 127
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Trees and Buildings
Volumes
Complements of compacta
µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn
We compute volumes of cusps directly
Using an explicit description in terms of algebraic data attached to G
Building on work of Harder, Soule and others
122 / 127
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Trees and Buildings
Volumes
Complements of compacta
µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn
We compute volumes of cusps directly
Using an explicit description in terms of algebraic data attached to G
Building on work of Harder, Soule and others
123 / 127
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Trees and Buildings
Volumes
Complements of compacta
µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn
We compute volumes of cusps directly
Using an explicit description in terms of algebraic data attached to G
Building on work of Harder, Soule and others
124 / 127
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Trees and Buildings
Volumes
Complements of compacta
µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn
We compute volumes of cusps directly
Using an explicit description in terms of algebraic data attached to G
Building on work of Harder, Soule and others
125 / 127
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Trees and Buildings
Picture credits
D. Sullivan, Acta Math 1982
Ulrich Gortz, Modular forms and special cycles on Shimura curves, S.Kudla, M. Rapoport and T. Yang
126 / 127
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Trees and Buildings
Picture credits
D. Sullivan, Acta Math 1982
Ulrich Gortz, Modular forms and special cycles on Shimura curves, S.Kudla, M. Rapoport and T. Yang
127 / 127