## Notes of Mikolai Fraczyk’s Cambridge lecture 19-04-2017

Benjamin-Schramm convergence of arithmetic 3-manifolds

1. B-S convergence and limit multiplicity

Let ${G}$ be a semi-simple Lie group, let ${\Gamma_i}$ be a sequence of lattices in ${G}$, without infinite repetitions. We say that locally symmetric spaces ${\Gamma_i\setminus X}$ B-S converge if the relative volume of the thin part tends to 0. The limit is then a probability distribution on the set (topologized by Gromov-Hausdorff convergence) of finite volume locally symmetric spaces.

In their 7-author paper, Abert-Bergeron-… showed that

1. If ${G}$ has higher rank ans has property (T), then for every non-trivial sequence, the limit is ${X}$.
2. For general semisimple ${G}$, given an arithmetic lattice, the relative volume of the thin parts of its congruence coverings is at most a negative power of the volume. Thus B-S convergence holds.

1.1. Limit multiplicity

If ${\Gamma is a lattice, ${G}$ acts on ${L^2(\Gamma\setminus G)}$. If ${\Gamma}$ is uniform, this unitary representation ${R_\Gamma}$ splits as a direct sum of irreducibles ${\pi\in \hat G}$ with multiplicities ${m_\Gamma(\pi)}$. We define the atomic measure on ${\hat G}$

$\displaystyle \begin{array}{rcl} \mu_\Gamma:=\frac{1}{\mathrm{Vol}(\Gamma\setminus G)}\sum_{\pi\in \hat G}m_\Gamma(\pi)\delta_\pi, \end{array}$

Definition 1 Let ${(\Gamma_i)}$ be a sequence of uniform lattices in ${G}$. Say that it has the limiting multiplicity property if ${\mu_{\Gamma_i}}$ converges to the Plancherel measure on ${\hat G}$.

Sauvageot (1996) gave the following characterization. L-M property holds iff for every smooth compactly supported function ${f}$ on ${G}$,

$\displaystyle \begin{array}{rcl} \lim_{i\rightarrow\infty}\frac{\mathrm{Trace}(R_{\Gamma_i}(f)}{\mathrm{Vol}(\Gamma_i\setminus G)}=f(1). \end{array}$

Abert-Bergeron et al showed that if injectivity radii are uniformly bounded from below, B-S convergence implies limiting multiplicity property. The reason is that injectivity radius at the base-point tends to infinity, thus the effect of convolution with a small support kernel becomes independant of the kernel.

The converse is true (folklore): limiting multiplicity property implies B-S convergence.

2. Result

I complete the discussion of congruence lattices in the remaining 2 and 3-dimensional cases.

2.1. Torsion free uniform case

Theorem 2 Let ${G=PGl(2,{\mathbb R})}$ or ${PGl(2,{\mathbb C})}$. Let ${\Gamma}$ be a congruence uniform torsion free arithmetic lattice. Then

1. Thin parts have small volume,

$\displaystyle \begin{array}{rcl} \mathrm{Vol}((\Gamma\setminus G)_{

where constants depend only on ${R}$. Note that “congruence” is intrinsically defined.

2. Limiting multiplicity property holds. Given smooth compactly supported function ${f}$ on ${G}$ such that ${\|f\|_\infty\leq 1}$, support in ${B(1,R)}$, then

$\displaystyle \begin{array}{rcl} |\frac{\mathrm{Trace}(R_{\Gamma}(f))}{\mathrm{Vol}(\Gamma\setminus G)}-f(1)|<_R\mathrm{Vol}(\Gamma\setminus G)^{-\delta}. \end{array}$

3. In the non-congruence arithmetic case, the estimate on the thin part involves the discriminant of the trace field (the field generated by traces of matrices ${Ad_\gamma}$, ${\gamma\in\Gamma}$,

$\displaystyle \begin{array}{rcl} \mathrm{Vol}((\Gamma\setminus G)_{

In fact, (2), applied to a smoothed characteristic function of a ball implies (1).

Odlyzko’s bound indicates that discriminants are usually large:

$\displaystyle \begin{array}{rcl} \Delta(k)>60\#\{\textrm{real places}\}\times 22\#\{\textrm{complex places}\}. \end{array}$

Corollary 3 For ${\Gamma}$ arithmetic in ${Sl(2,{\mathbb C})}$, ${M=\Gamma\setminus H^3}$ admits a triangulation ${T}$ with ${O(\mathrm{Vol}(M))}$ vertices, degrees of vertices ${\leq 245}$.

This had been conjectured by Gelander. There are many small triangles in the thin part, but their number is overwhelmed by large triangles of the thick part. The corollary relies on Dobrowolski’s 1976 theorem on Mahler measures of algebaric numbers, which yields a lower bound on injectivity radius

$\displaystyle \begin{array}{rcl} injrad(M)>(\log[k(\Gamma):{\mathbb Q}])^{-3}. \end{array}$

${T}$ is obtained as the nerve of a cover of ${M}$ by balls.

2.2. Torsion case, non-uniform case

(Joint work with Jean Raimbault). We prove that every sequence of arithmetic congruence lattices of ${PGl(2,{\mathbb R})}$ or ${PGl(2,{\mathbb C})}$ B-S converges to ${X}$.

2.3. Proof of torsion case, non-uniform case

Follow the 7 author paper. Consider the invariant random subgroup ${\nu_\Gamma}$ associated to ${\Gamma}$ (uniform measure on conjugates in the Chabauty space of closed subgroups of ${G}$. The 7 show that B-S convergence is equivalent to convergence of the IRS to the Dirac measure at the trivial subgroup. We show that any limit of ${\nu_{\Gamma_i}}$ is supported on subgroups containing only torsion by unipotent elements. Such a subgroup cannot be Zariski dense. The 7 show that every nontrivial IRS is Zariski dense.

3. Proof of main theorem

The goal is the estimate

$\displaystyle \begin{array}{rcl} |\frac{\mathrm{Trace}(R_{\Gamma}(f))}{\mathrm{Vol}(\Gamma\setminus G)}-f(1)|<_R\mathrm{Vol}(\Gamma\setminus G)^{-\delta}. \end{array}$

Using Selberg’s trace formula,

$\displaystyle \begin{array}{rcl} \mathrm{Trace}(R_{\Gamma}(f))=\sum_{\mathrm{conjugacy\,classes}\,[\gamma]}\mathrm{Vol}(\Gamma_\gamma\setminus G_\gamma)\int_{G_\gamma\setminus G}f(x^{-1}\gamma x)\,dx. \end{array}$

The dominant term comes from the trivial conjugacy class, one must estimate all other terms. If support${(f)\subset B(1,R)}$, the number of nonzero terms is the number of closed geodesics of length ${, i.e. elements with eigenvalues ${< e^R}$. These eigenvalues ${\lambda}$ are algebraic integers of a special type. Indeed, uniform arithmetic lattices are integer matrices in a product of one copy of ${Sl(2)}$ and a number of orthogonal groups. Therefore, among the Galois conjugates of ${\lambda}$, at most two can have modulus ${\not=1}$. The Weil height ${h(\lambda)}$ is ${O(1/[{\mathbb Q}[\lambda]:{\mathbb Q}])}$, it is very small. Equidistribution results exist for numbers of small Weil height.

Theorem 4 (Bilu) If ${z_i}$ are algebraic numbers of degrees tending to infinity and Weil height tending to 0, then the Galois conjugates of ${z_i}$ get uniformly distributed on the unit circle.

This implies a bound on nonzero terms,

$\displaystyle \begin{array}{rcl} \mathrm{Vol}(\Gamma_\gamma\setminus G_\gamma)<\Delta_{k(\Gamma)}^{1/2}e^{o([k(\Gamma):{\mathbb Q}])}, \end{array}$

$\displaystyle \begin{array}{rcl} |\int_{G_\gamma\setminus G}f(x^{-1}\gamma x)\,dx|

and also on the number nonzero terms, when combined with a result of Kabatianskii-Levenstein on the number of nearly orthogonal unit vectors in Euclidean space.

Theorem 5 (Kabatianskii-Levenstein, Tao) Let ${v_1,\ldots,v_m\in {\mathbb R}^n}$ be unit vectors that satisfy

$\displaystyle \begin{array}{rcl} |v_i\cdot v_j|\leq \frac{A}{n}. \end{array}$

Then ${m\ll n^{CA}}$, for some absolute constant ${C}$.

Indeed, one constructs a unit vector in ${{\mathbb R}^{[k(\Gamma):{\mathbb Q}]-2}}$ for each characteristic polynomial of element of length ${. Bilu’s theorem implies that they are nearly orthogonal.

Borel’s volume formula gives

$\displaystyle \begin{array}{rcl} \mathrm{Vol}(\Gamma\setminus G) \sim \Delta_{k(\Gamma)}^{3/2}\frac{\zeta_{k(\Gamma)}(2)}{(4\pi^2)^{[k(\Gamma):{\mathbb Q}]}}> \Delta_{k(\Gamma)}^{1/2+\epsilon} \end{array}$

thanks to Odlyzko’s bound.

This gives the result for maximal lattices. Generalizing to suitable finite index subgroups requires the notion of congruence subgroup.

Let ${A}$ be a quaternion algebra over a number field ${k}$. Let ${G}$ be the group of unit quaternions. Then ${G(k\otimes{\mathbb R})}$ is a product of ${Sl(2,{\mathbb R})}$‘s, ${Sl(2,{\mathbb C})}$‘s and ${SO(3)}$‘s. Adjusting parameters allows to have exactly one noncompact factor. Pick an order ${\mathcal{O}}$ of ${A}$, i.e. an ${\mathcal{O}_k}$-submodule which is a subring. Then ${\mathcal{O}\cap G}$ is a lattice in ${G(k\otimes{\mathbb R})}$, which projects to a lattice of ${Sl(2)}$. The previous arguments apply to such lattices, provided ${\mathcal{O}}$ is maximal. Say a lattice ${\Gamma}$ in ${Sl(2)}$ is congruence if it contains the projection of ${\{x\in\mathcal{O}\cap G\,;\, x\equiv 1\mod N\mathcal{O}\}}$ for some order ${\mathcal{O}}$ and some integer ${N}$.

To treat the case of a congruence lattice ${\Gamma<\Gamma_{max}}$, one views ${L^2(\Gamma\setminus G)}$ as sections of a vectorbundle over ${\Gamma_{max}\setminus G}$ associated with representations of ${Sl(2,{\mathbb Z}/N{\mathbb Z})}$.