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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


Heirarchy

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Shrinking

Target

Problems

0− 1 laws


Heirarchy

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Shrinking

Target

Problems

0− 1 laws


Heirarchy

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Shrinking

Target

Problems

0− 1 laws


Heirarchy

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Shrinking

Target

Problems

0− 1 laws


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)




if and only if

∞∑t=1

e−krt =∞

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Sullivan (1982)




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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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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‖Aq + p‖m < ψ(‖q‖n)

if and only if

∞∑t=1

ψ(t) =∞

If time permits


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‖Aq + p‖m < ψ(‖q‖n)

if and only if

∞∑t=1

ψ(t) =∞

If time permits


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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

)



is bounded

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Dani Correspondence

x ! ux :=

(1 x0 1

)

gt :=

(et 00 e−t

)



is bounded

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Dani Correspondence

x ! ux :=

(1 x0 1

)

gt :=

(et 00 e−t

)



is bounded

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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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∞∑t=0

vol(Xn(t))

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∞∑t=0

vol(Xn(t))

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∞∑t=0

vol(Xn(t))

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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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|〈gφ, ψ〉| � L2l (φ)L2

l (ψ)H(g)−λ


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|〈gφ, ψ〉| � L2l (φ)L2

l (ψ)H(g)−λ


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|〈gφ, ψ〉| � L2l (φ)L2

l (ψ)H(g)−λ


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|〈gφ, ψ〉| � L2l (φ)L2

l (ψ)H(g)−λ


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|〈gφ, ψ〉| � L2l (φ)L2

l (ψ)H(g)−λ


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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 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





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Key technical point





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Key technical point





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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






For g , h ∈ S

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Volume estimates






For g , h ∈ S

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Volume estimates






For g , h ∈ S

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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)

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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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Local fields of positive characteristic

Fp - finite field

Fp[X ] ! Z

Fp(X ) ! Q

Fp((X−1)) ! R

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Fp - finite field

Fp[X ] ! Z

Fp(X ) ! Q

Fp((X−1)) ! R

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Fp - finite field

Fp[X ] ! Z

Fp(X ) ! Q

Fp((X−1)) ! R

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Trees and Buildings


Fp - finite field

Fp[X ] ! Z

Fp(X ) ! Q

Fp((X−1)) ! R

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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






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Euclidean buildings

G has a BN pair






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Trees and Buildings

Euclidean buildings

G has a BN pair






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Euclidean buildings

G has a BN pair






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Euclidean buildings

G has a BN pair






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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.

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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 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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Trees continued





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Trees continued





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Trees continued





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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

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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

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And their quotients





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And their quotients





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And their quotients





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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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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

∆ 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

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∆ 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

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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

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supm∈N

∞∑n=1

‖fnf −1m ‖−β <∞ ∀ β > 0,


B(∆) := {{x ∈ G/Γ | ∆(x) ≥ sn} | n ∈ Z}


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supm∈N

∞∑n=1

‖fnf −1m ‖−β <∞ ∀ β > 0,


B(∆) := {{x ∈ G/Γ | ∆(x) ≥ sn} | n ∈ Z}


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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

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Ergodic ideas






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Ergodic ideas






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Ergodic ideas






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Ergodic ideas






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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

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Effective mixing


0(G/Γ),

|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.


Clozel and Ullmo


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Effective mixing


0(G/Γ),

|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.


Clozel and Ullmo


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Effective mixing


0(G/Γ),

|〈ρ0(g)φ, ψ〉| � ‖φ‖2‖ψ‖2H(g)−1/b.


Clozel and Ullmo


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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

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Volumes


µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn




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Volumes


µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn




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Volumes


µ({x ∈ G/Γ : d(x0, x) ≥ sn}) � s−κn




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Picture credits

D. Sullivan, Acta Math 1982

Ulrich Gortz, Modular forms and special cycles on Shimura curves, S.Kudla, M. Rapoport and T. Yang

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Picture credits

D. Sullivan, Acta Math 1982

Ulrich Gortz, Modular forms and special cycles on Shimura curves, S.Kudla, M. Rapoport and T. Yang

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