Notes of Frederic Paulin’s lecture

Equidistribution in the Heisenberg group

Joint with J. Parkkonen.

I view the Heisenberg group as a real quadric

\displaystyle  \begin{array}{rcl}  Heis_3=\{(w_0,w)\in{\mathbb C}\times{\mathbb C}\,;\,2\Re e(w_0)=|w|^2\}. \end{array}

In these coordinates, Haar measure is {d(\Im m(w_0))\frac{dwd\bar{w}}{i}}. The map {(w_0,w)\mapsto w}, {Heis_3\rightarrow{\mathbb C}}, is a group homomorphism.

1. Equidistribution of rational points

The field {{\mathbb Q}(i)} will play the role of the rationals, and {\mathcal{O}={\mathbb Z}[i]} the role of integers. requiring that {w_0} and {w} belong to {K} defines the rational Heisenberg group {Heis_3({\mathbb Q})}. Elements of {Heis_3({\mathbb Q})} can be uniquely written as fractions {(\frac{a}{c},\frac{b}{c})} with mutually prime Gaussian integers {a}, {b}, {c\in{\mathbb Z}[i]}.

Theorem 1 At rational points of denominator of modulus {<s} (the number of them in a ball grows like {s^{-4}}), put a unit Dirac mass. Then the normalized sum weakly converges to Haar measure up to the multiplicative constant

\displaystyle  \begin{array}{rcl}  c_1=2\pi\frac{|D_k|^{3/2}\zeta_K(3)}{\zeta(3)}. \end{array}

Corollary 2 Modulo the integer points, {Heis_3({\mathbb Z})}, The numbder of rational points of denominator {<s} is {\sim \frac{1}{c_1}s^4}.

2. Counting arithmetic chains

2.1. Chains

Let {h=-z_0\bar{z}_2-z_2\bar{z}_0+|z_1|^2} be the signature {(1,2)} Hermitian form in {{\mathbb C}^3}. The Poincaré hypersphere

\displaystyle  \begin{array}{rcl}  \mathcal{HS}\{[z_0:z_1:z_2]\in P^2({\mathbb C})\,;\,h(z_0,z_1,z_2)=0\}. \end{array}

is a compactification of {Heis_3}. Indeed, {Heis_3} embeds in it via {(w_0,w)\mapsto[1:w:w_0]} as the complement of single point {\infty=[0:0:1]}.

On {Heis_3} we shall use the Cygan distance (aka Koranyi distance). It is the unique left-invariant on {Heis_3} such that

\displaystyle  \begin{array}{rcl}  d_{Cyg}((w_0,w),(0,0))=\sqrt{2|w_0|}. \end{array}

And also a modified Cygan distance

\displaystyle  \begin{array}{rcl}  d''_{Cyg}((w_0,w),(0,0))=\frac{2|w_0|}{\sqrt{|w|^2+2|w_0|}}. \end{array}

It is equivalent to {d_{Cyg}} or the a Carnot-Caratheodory metric. We need this exact expression in order to state sharp estimates.

Definition 3 (von Staudt) A chain is the intersection of Poincaré’s hypersphere with a projective line.

In {Heis_3}, those which contain {\infty} are lines {\{w=const.\}}. The others are ellipses which projet to circles in {{\mathbb C}}.

2.2. Arithmetic groups and chains

{SL(3,{\mathbb C})} on {P^2({\mathbb C})} and preserves lines. The subgroup {SU_h} preserves the Hermitian form {h} and therefore acts on Poincarés hypersphere. Let

\displaystyle  \begin{array}{rcl}  \Gamma=SU_h \cap SL_3({\mathbb Z}[i]). \end{array}

This is a discrete subgroup of {SU_h}, which preserves chains.

Definition 4 A chain is arithmetic if its stabilizer in {\Gamma} has a dense orbit in the chain.

Theorem 5 Let {C_0} be an arithmetic chain. There exist constants {\kappa>0} and {c_2>0} (depending on {C_0}) such that

\displaystyle  \begin{array}{rcl}  |Heis_3({\mathbb Z})\setminus\{c\in\Gamma C_0\,;\, \mathrm{diameter}_{d''_{Cyg}}(c)\geq\epsilon\}|\sim c_2 \epsilon^{-4}(1+O(\epsilon^\kappa)). \end{array}

3. Complex hyperbolic geometry

This enters the proof. Complex hyperbolic space has curvature between {-4} and {-1}. We use the Siegel domain model

\displaystyle  \begin{array}{rcl}  \{(w_0,w)\in{\mathbb C}\,;\, 2\Re e(w_0)-|w|^2>0\} \end{array}

and the projective model

\displaystyle  \begin{array}{rcl}  \{[z_0:z_1:z_2]\in P^2({\mathbb C})\,;\, h(z_0,z_1,z_2)<0\} \end{array}

related by the map {(w_0,w)\mapsto [1:w:w_0]}. Taking closures, we recover the previously encountered {Heis_3} and {\mathcal{HS}}. The identity component of their isometry group is {PSU_h}. The subset

\displaystyle  \begin{array}{rcl}  \{(w_0,w)\in{\mathbb C}\,;\, 2\Re e(w_0)-|w|^2>1\} \end{array}

is a horoball centered at {\infty}. {PSU_h} is transitive on horoballs.

Definition 6 A complex geodesic is the intersection of {H^2({\mathbb C})} with a projective line.

Complex geodesics are totally geodesic with curvature {-4}. Chains {c} coincide with ideal boundaries of complex geodesics {L}. A chain {c} is arithmetic if and only if {L} is, i.e. the stabilizer of {L} in {\Gamma} has finite covolume on {L}.

4. Equidistribution of common perpendiculars

Fix an arithmetic complex geodesic {L_0} with ideal boundary {C_0}. We project {H^2({\mathbb C})} radially from {\infty} onto {Heis_3} via

\displaystyle  \begin{array}{rcl}  (w_0,w)\mapsto(\frac{1}{2}|w|^2+i\Im m(w_0),w). \end{array}

This allows to identify the boundary of the model horoball {H} with {Heis_3}.

Consider common perpendiculars

  • of {H} and {H'}, horoballs.
  • of {H} and a complex geodesic.

I both cases, these are vertical segments.

Theorem 7 Put a unit Dirac mass at each footpoint {x_\gamma} of a common perpendicular of {H} with {\gamma H} whose height is {>t}. Renormalize by {e^{-4t}}. Then the resulting sum weakly converges to Haar measure on {Heis_3} up to a mltiplicative constant

\displaystyle  \begin{array}{rcl}  c_3=2\pi\frac{vol()vol()}{vol()}. \end{array}

The values of {\zeta} functions in previous theorem arive from these volumes.

4.1. Proof

Use a formula for the Poisson measure and the measure of maximal entropy in terms of the Cygan distance. Then establish exponential decay of correlations. Here is the crucial Lemma.

Lemma 8

\displaystyle  \begin{array}{rcl}  d(H,\gamma H)=2\log|c| \end{array}

if the point at infinity of {\gamma H} is {(\frac{a}{c},\frac{b}{c})}.

\displaystyle  \begin{array}{rcl}  d(H,\gamma L_0)=-\log(\frac{\mathrm{diameter}(\gamma C_0)}{\sqrt{2}}). \end{array}

whence the change of variable {s=e^t}.

Next seminar on may 14th.


About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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