Geodesic flows on quotients of Bruhat-Tits
buildings
and Number Theory
Anish Ghosh
University of East Anglia
joint with J. Athreya (Urbana) and A. Prasad (Chennai)
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Diagonal Actions
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Diagonal Actions
Recurrence
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Diagonal Actions
Recurrence
Buildings
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Diagonal Actions
Recurrence
Buildings
Shrinking Targets
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Shrinking
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Shrinking
Target
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Shrinking
Target
Problems
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Shrinking
Target
Problems
0 − 1 laws
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Shrinking
Target
Problems
0 − 1 laws
Return time asymptotics
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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
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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 many
t ∈ N such that
dist(y0 , γt (y , ξ)) ≥ rt
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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 many
t ∈ N such that
dist(y0 , γt (y , ξ)) ≥ rt
if and only if
∞
X
e −krt = ∞
t=1
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Logarithm Laws
Corollary
For almost every y ∈ Y and almost all ξ ∈ Ty (Y )
lim sup
t→∞
1
dist(y , γt (y , ξ))
=
log t
k
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Some Number theory
√
For almost every z, there exist infinitely many pairs (p, q) ∈ O( −d)
such that
√
p
ψ(|q|)
|z − | <
and (p, q) = O( −d)
2
q
|q|
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Some Number theory
√
For almost every z, there exist infinitely many pairs (p, q) ∈ O( −d)
such that
√
p
ψ(|q|)
|z − | <
and (p, q) = O( −d)
2
q
|q|
if and only if
∞
X
ψ(t)
t=1
t2
=∞
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition of
Margulis)
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition of
Margulis)
Along with geometric arguments
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition of
Margulis)
Along with geometric arguments
To force quasi-independence
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Sullivan’s strategy
Use the mixing property of the geodesic flow (in the tradition of
Margulis)
Along with geometric arguments
To force quasi-independence
And then apply the Borel-Cantelli Lemma
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γ
non-uniform lattice
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γ
non-uniform lattice
Y = K \G /Γ a locally symmetric space
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Kleinbock-Margulis (1999)
Generalized Sullivan’s results
G semisimple Lie group, K maximal compact subgroup, Γ
non-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 many
q ∈ Zn , p ∈ Zm such that
kAq + pkm < ψ(kqkn )
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Khintchine-Groshev Theorem
For almost all A ∈ Matm×n (R), there exist infinitely many
q ∈ Zn , p ∈ Zm such that
kAq + pkm < ψ(kqkn )
if and only if
∞
X
ψ(t) = ∞
t=1
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Khintchine-Groshev Theorem
For almost all A ∈ Matm×n (R), there exist infinitely many
q ∈ Zn , p ∈ Zm such that
kAq + pkm < ψ(kqkn )
if and only if
∞
X
ψ(t) = ∞
t=1
If time permits
More general multiplicative results
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SLn (R)/ SLn (Z) can be identified with the space Xn of unimodular
lattices in Rn
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SLn (R)/ SLn (Z) can be identified with the space Xn of unimodular
lattices in Rn
Mahler’s compactness criterion describes compact subsets of Xn
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SLn (R)/ SLn (Z) can be identified with the space Xn of unimodular
lattices in Rn
Mahler’s compactness criterion describes compact subsets of Xn
The systole of a lattice
s(Λ) := min kv k
v ∈Λ
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Dani Correspondence
x ! ux :=
1 x
0 1
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Dani Correspondence
x ! ux :=
gt :=
et
0
1 x
0 1
0
e −t
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Dani Correspondence
x ! ux :=
gt :=
et
0
1 x
0 1
0
e −t
Read Diophantine properties of x from the gt (semi) orbits of ux Z2
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Dani Correspondence
x ! ux :=
gt :=
et
0
1 x
0 1
0
e −t
Read Diophantine properties of x from the gt (semi) orbits of ux Z2
For example, x is badly approximable if and only if the orbit of ux Z2
is bounded
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More generally
There are infinitely many solutions to
|qx + p| < ψ(q)
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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(gt ux Z2 ) ≤ r (t)
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r (t) defines the complement Xn (t) of a compact set
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r (t) defines the complement Xn (t) of a compact set
Shrinking systolic neighborhoods
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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
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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
∞
X
vol(Xn (t))
t=0
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Similar story for logarithm laws
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Similar story for logarithm laws
Model geodesic flow on T 1 (K \G /Γ) by the action of diagonal
one-parameter subgroups on G /Γ
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Similar story for logarithm laws
Model geodesic flow on T 1 (K \G /Γ) by the action of diagonal
one-parameter subgroups on G /Γ
Compute vol({x ∈ G /Γ : d(x0 , gt x) ≥ r (t)})
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Similar story for logarithm laws
Model geodesic flow on T 1 (K \G /Γ) by the action of diagonal
one-parameter subgroups on G /Γ
Compute vol({x ∈ G /Γ : d(x0 , gt x) ≥ r (t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients for
automorphic representation
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Similar story for logarithm laws
Model geodesic flow on T 1 (K \G /Γ) by the action of diagonal
one-parameter subgroups on G /Γ
Compute vol({x ∈ G /Γ : d(x0 , gt x) ≥ r (t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients for
automorphic representation
For φ, ψ ∈ L2 (G /Γ), and g ∈ G
|hg φ, ψi| L2l (φ)L2l (ψ)H(g )−λ
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Similar story for logarithm laws
Model geodesic flow on T 1 (K \G /Γ) by the action of diagonal
one-parameter subgroups on G /Γ
Compute vol({x ∈ G /Γ : d(x0 , gt x) ≥ r (t)})
Use quantitative mixing, i.e. effective decay of matrix coefficients for
automorphic representation
For φ, ψ ∈ L2 (G /Γ), and g ∈ G
|hg φ, ψi| L2l (φ)L2l (ψ)H(g )−λ
To force quasi-independence on average
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Much recent activity
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Much recent activity
Maucourant, Hersonsky-Paulin, Parkonnen-Paulin, Chaika
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Much recent activity
Maucourant, Hersonsky-Paulin, Parkonnen-Paulin, Chaika
Nogueira, Laurent-Nogueira, G-Gorodnik-Nevo
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Key technical point
In order to apply decay of matrix coefficients we have to smoothen
cusp neighborhoods
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Key technical point
In order to apply decay of matrix coefficients we have to smoothen
cusp neighborhoods
And use suitable Sobolev norms
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Key technical point
In order to apply decay of matrix coefficients we have to smoothen
cusp neighborhoods
And use suitable Sobolev norms
This destroys sensitive information about exponents
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Key technical point
In order to apply decay of matrix coefficients we have to smoothen
cusp neighborhoods
And use suitable Sobolev norms
This destroys sensitive information about exponents
Which is crucial in G-Gorodnik-Nevo
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
¯ metric on G /Γ defined by
dist metric on G , dist
¯ Γ, hΓ) := inf {dist(g γ, h) : γ ∈ Γ}
dist(g
γ∈Γ
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
¯ metric on G /Γ defined by
dist metric on G , dist
¯ Γ, hΓ) := inf {dist(g γ, h) : γ ∈ Γ}
dist(g
γ∈Γ
Coarse isometry (Leuzinger + Ji)
¯ Γ, hΓ) ≤ dist(g , h) ≤ dist(g
¯ Γ, hΓ) + C
dist(g
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Volume estimates
Γ arithmetic lattice in G , S Siegel set for Γ in G
¯ metric on G /Γ defined by
dist metric on G , dist
¯ Γ, hΓ) := inf {dist(g γ, h) : γ ∈ Γ}
dist(g
γ∈Γ
Coarse isometry (Leuzinger + Ji)
¯ Γ, hΓ) ≤ dist(g , h) ≤ dist(g
¯ Γ, hΓ) + C
dist(g
For g , h ∈ S
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Trees and Buildings
If G is a p-adic Lie group then
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If G is a p-adic Lie group then
The analogue of a symmetric space is a Bruhat-Tits building
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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
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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 )
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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
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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
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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)
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However, we could consider G = PGL2 (Fp ((t))) × PGL2 (Fp ((t −1 )))
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However, we could consider G = PGL2 (Fp ((t))) × PGL2 (Fp ((t −1 )))
And Γ = PGL2 (Fp [t, t −1 ])
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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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The Tree of SL2 (F2 ((X −1 )))
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Ultrametric logarithm laws
So it makes sense to investigate analogues of logarithm laws for
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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
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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
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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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Local fields of positive characteristic
Fp - finite field
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Local fields of positive characteristic
Fp - finite field
Fp [X ] ! Z
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Local fields of positive characteristic
Fp - finite field
Fp [X ] ! Z
Fp (X ) ! Q
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Local fields of positive characteristic
Fp - finite field
Fp [X ] ! Z
Fp (X ) ! Q
Fp ((X −1 )) ! R
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Euclidean buildings
G has a BN pair
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Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
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Euclidean buildings
G has a BN pair
Corresponding to which there is a Euclidean building
A Euclidean building is polyhedral complex
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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
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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
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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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Borel-Harish-Chandra, Behr-Harder
G (Fp [X ]) is a lattice in G (Fp ((X −1 )))
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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-arithmetic
lattices.
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Logarithm laws for trees
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Logarithm laws for trees
T - a regular tree, then Aut(T ) is a locally compact group
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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
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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 )
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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
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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
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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 continued
They also obtained generalizations of Khintchine’s theorem
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Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
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Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
Their method is very geometric
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Trees continued
They also obtained generalizations of Khintchine’s theorem
In positive characteristic
Their method is very geometric
And reminiscent of Sullivan
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Let G be a connected, semisimple linear algebraic group defined and
split over Fp
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Let G be a connected, semisimple linear algebraic group defined and
split over Fp
Let Γ be an S arithmetic lattice in GS
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Let G be a connected, semisimple linear algebraic group defined and
split over Fp
Let Γ be an S arithmetic lattice in GS
By a result of Margulis + Venkataramana
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Let G be a connected, semisimple linear algebraic group defined and
split 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
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And their quotients
We prove analogues of the results of Kleinbock-Margulis
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And their quotients
We prove analogues of the results of Kleinbock-Margulis
Namely logarithm laws for arithmetic quotients of
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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
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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
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Main Result
Let B be a family of measurable subsets and F = {fn } denote a
sequence of µ-preserving transformations of G /Γ
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Main Result
Let B be a family of measurable subsets and F = {fn } denote a
sequence of µ-preserving transformations of G /Γ
B is Borel-Cantelli for F if for every sequence {An : n ∈ N} of sets
from B
µ({x ∈ G /Γ | fn (x) ∈ An for infinitely many n ∈ N})

P∞

0 if
n=1 µ(An ) < ∞
=

P∞

1 if
n=1 µ(An ) = ∞
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∆ a function on G /Γ define the tail distribution function
Φ∆ (n) := µ({x ∈ G /Γ | ∆(x) ≥ s n })
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∆ a function on G /Γ define the tail distribution function
Φ∆ (n) := µ({x ∈ G /Γ | ∆(x) ≥ s n })
Call ∆ “distance-like” if
Φ∆ (n) s −κn ∀ n ∈ Z
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Let F = {fn | n ∈ N} be a sequence of elements of G satisfying
sup
∞
X
m∈N n=1
kfn fm−1 k−β < ∞ ∀ β > 0,
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Let F = {fn | n ∈ N} be a sequence of elements of G satisfying
sup
∞
X
m∈N n=1
kfn fm−1 k−β < ∞ ∀ β > 0,
and let ∆ be a “distance-like” function on G /Γ. Then
B(∆) := {{x ∈ G /Γ | ∆(x) ≥ s n } | n ∈ Z}
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Let F = {fn | n ∈ N} be a sequence of elements of G satisfying
sup
∞
X
m∈N n=1
kfn fm−1 k−β < ∞ ∀ β > 0,
and let ∆ be a “distance-like” function on G /Γ. Then
B(∆) := {{x ∈ G /Γ | ∆(x) ≥ s n } | n ∈ Z}
is Borel-Cantelli for F
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Ergodic ideas
We establish effective decay of matrix coefficients for semisimple
groups in positive characteristic
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Ergodic ideas
We establish effective decay of matrix coefficients for semisimple
groups in positive characteristic
Well known to experts but unavailable in the literature
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Ergodic ideas
We establish effective decay of matrix coefficients for semisimple
groups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regular
representation of G on G /Γ
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Ergodic ideas
We establish effective decay of matrix coefficients for semisimple
groups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regular
representation of G on G /Γ
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Ergodic ideas
We establish effective decay of matrix coefficients for semisimple
groups in positive characteristic
Well known to experts but unavailable in the literature
Main idea is to establish strong spectral gap for the regular
representation of G on G /Γ
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Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined and
quasi-split over k and Γ be an non-uniform irreducible lattice in G . For
any g ∈ G and any smooth functions φ, ψ ∈ L20 (G /Γ),
|hρ0 (g )φ, ψi| kφk2 kψk2 H(g )−1/b .
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Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined and
quasi-split over k and Γ be an non-uniform irreducible lattice in G . For
any g ∈ G and any smooth functions φ, ψ ∈ L20 (G /Γ),
|hρ0 (g )φ, ψi| kφk2 kψk2 H(g )−1/b .
We use the Burger-Sarnak method as extended to p-adic and positive
characteristic fields by
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Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined and
quasi-split over k and Γ be an non-uniform irreducible lattice in G . For
any g ∈ G and any smooth functions φ, ψ ∈ L20 (G /Γ),
|hρ0 (g )φ, ψi| kφk2 kψk2 H(g )−1/b .
We use the Burger-Sarnak method as extended to p-adic and positive
characteristic fields by
Clozel and Ullmo
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Effective mixing
Let G be a connected, semisimple, linear algebraic group, defined and
quasi-split over k and Γ be an non-uniform irreducible lattice in G . For
any g ∈ G and any smooth functions φ, ψ ∈ L20 (G /Γ),
|hρ0 (g )φ, ψi| kφk2 kψk2 H(g )−1/b .
We use the Burger-Sarnak method as extended to p-adic and positive
characteristic fields by
Clozel and Ullmo
Along with the solution of the Ramanujan conjecture for GL2 due to
Drinfield
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Volumes
Complements of compacta
µ({x ∈ G /Γ : d(x0 , x) ≥ s n }) s −κn
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Volumes
Complements of compacta
µ({x ∈ G /Γ : d(x0 , x) ≥ s n }) s −κn
We compute volumes of cusps directly
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Volumes
Complements of compacta
µ({x ∈ G /Γ : d(x0 , x) ≥ s n }) s −κn
We compute volumes of cusps directly
Using an explicit description in terms of algebraic data attached to G
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Volumes
Complements of compacta
µ({x ∈ G /Γ : d(x0 , x) ≥ s n }) 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, Soulé and others
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Picture credits
D. Sullivan, Acta Math 1982
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Picture credits
D. Sullivan, Acta Math 1982
Ulrich Görtz, Modular forms and special cycles on Shimura curves, S.
Kudla, M. Rapoport and T. Yang
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