Uniform Tits Alternative II: local and global bounds on the height gap

This is the second post about Breuillard's proof of the uniform Tits alternative. As mentioned last time we will mostly be interested in the Height Gap Theorem.

Uniform Height Gap Theorem: There is a constant $\varepsilon =\varepsilon (d)>0$ such that if $F$ is a finite subset of $GL_{d}(\overline{\mathbb{Q}})$ generating a non amenable subgroup that acts strongly irreducibly, then $\widehat{h} (F)>\varepsilon $.

Last time we defined heights and mentioned their basic properties. In this post we finish our discussion of the properties that we will need. (the numbering will also continue from last time) The strategy for bounding them is divide and conquer: for every place, Breuillard uses a string of Lie algebraic techniques to relate the minimal norm $E_k(F)$ and the matrix coefficients of the elements of $F$ in the adjoint representation. In particular, he gives bounds on various matrix norms. We leave it to the interested reader to look into Breuillard's paper for details. Having these bounds for every place, we then apply all of them to control the behaviour of the height function globally. I will also be brief in this part because the calculation is, to me at least, just a calculation. First, some (a lot of) notation.

Denote by $\mathbb{G}$ a Chevalley group of adjoint type and $T$ a maximal torus. We fix a total order on the set of all roots induced by an ordering of the simple roots, that is $\Phi =\{\alpha _{1},...,\alpha _{|\Phi ^{+}|},\alpha _{|\Phi ^{+}|+r+1},...,\alpha _{d}\}$, where $r$ is the rank of $\mathfrak{g}=Lie(\mathbb{G})$ and $I_{r}=[|\Phi ^{+}|,|\Phi ^{+}|+r]$.
The Lie algebra $\mathfrak{g}$ has a basis $(Y_{1},...,Y_{d})$ obtained from a Chevalley basis of $\mathfrak{g}$, with $Y_{i}=X_{\alpha _{i}}$ if $i\notin I_{r}$ and $Y_{i}\in \{\omega _{\alpha },\alpha \in \Pi \}$ if $i\in I_{r}.$ Also $\mathfrak{g}_{\mathbb{Z}}$ denotes the integer lattice generated by the basis $(Y_{1},...,Y_{d}).$ For $X,Y\in \mathfrak{g}$ we set $\phi (X,Y)=-B(X^{\tau },Y)$ where $B$ is the Killing form and $\tau $ the Chevalley involution.
For $\alpha \in \Phi $, let $\mathfrak{g}_{\alpha }$ be the root subspace corresponding to $\alpha $ and $\mathfrak{t}=\mathfrak{g}_{0}$ be the Lie algebra of $T,$ so that we have the direct sum decomposition

\[\mathfrak{g}=\mathfrak{t}\oplus \bigoplus_{\alpha \in \Phi }\mathfrak{g}_{\alpha }  \]

which is an orthogonal decomposition for the symmetric bilinear form $\phi $.

We will consider the elements $A=Ad(a)$ and $B=Ad(b)$ from $F=\{Id,a,b\}\subset \mathbb{G}(\overline{\mathbb{Q}})$ with $a\in T$ as
matrices in the basis $(Y_{1},...,Y_{d})$. Then $A$ is diagonal and $B=(b_{ij})_{ij}\in SL_{d}(\overline{\mathbb{Q}})$. Consider the regular
function on $\mathbb{G}$ given by $f(g)=g_{dd}$ in this basis. The root $\alpha \left( d\right) $ is the smallest root in the above ordering. It coincides with the opposite of the highest root of $\Phi $.Observe the following:

$-$ for every $t\in T$, we have $f(tgt^{-1})=f(g).$

$-$ for every $t\in T$, $f(t)=\alpha _{d}(t)$, hence $f$ is not constant.

$-$ $\phi (Ad(g)Y_{d},Y_{d})=f(g)\phi (Y_{d},Y_{d}),$

$-$ for every place $v$ we have $|f(g)|_{v}\leq ||Ad(g)||_{v}$.

Recall that we consider $\mathbb{G}$ as a subgroup of $SL(\mathfrak{g})$ and thus define the heights $e$ and $\widehat{h}$ of finite subsets of $\mathbb{G(}\overline{\mathbb{Q}})$ with respect to the adjoint representation. The important result is:

Proposition 11: For every $n\in \mathbb{N}$ and any $\alpha >0$ there is $\eta >0$ and $A_{1}>0$ such that if $F=\{Id,a,b\}$ is a subset of $\mathbb{G} (\overline{\mathbb{Q}})$ with $a\in T(\overline{\mathbb{Q}})$ such that $e(F)<\eta $ and $\deg (\alpha _{i}(a))>A_{1}$ for each positive root $\alpha_{i},$ then we have for every $i\in \mathbb{N}$, $1\leq i\leq n$

\[h(f(b^{i}))<\alpha \]
where $f$ is the function defined above.

The main extra ingredient needed for the proof, apart from the results local at every place, is this:
 

Bilu's Equidistribution Theorem of Small Points Suppose $(\lambda _{n})_{n\geq 1}$ is a sequence of algebraic numbers (i.e. in $\overline{\mathbb{Q}}$) such that $h(\lambda _{n})\rightarrow 0$ and $\deg(\lambda _{n})\rightarrow +\infty $ as $n\rightarrow +\infty .$ Let $\mathcal{O}(\lambda _{n})$ be the Galois orbit of $\lambda _{n}$. Then we have the following weak-$*$ convergence of probability measures on $\mathbb{C}$,

\[\frac{1}{\#\mathcal{O}(\lambda _{n})}\sum_{x\in \mathcal{O}(\lambda_{n})}\delta _{x}\underset{n\rightarrow +\infty }{\rightarrow }d\theta \]

where $d\theta $ is the normalized Lebesgue measure on the unit circle $\{z\in \mathbb{C}$, $|z|=1\}.$

The reason this is needed is that the upper bounds for various matrix norms is in terms of an average of $\log|1-\alpha(A)|_v$ over the Archimedean places. If the Galois conjugates equidistribute on the unit circle, the average becomes small. We illustrate how to use this to get bounds on heights with the next lemma. For $A\geq 1$ and $x\in \overline{\mathbb{Q}}$ we set

\[h_{\infty }^{A}(x):=\frac{1}{[K:\mathbb{Q}]}\sum_{v\in V_{\infty},|x|_{v}\geq A}n_{v}\cdot \log ^{+}|x|_{v}  \]
where the sum is limited to those $v\in V_{\infty }$ such that $|x|_{v}\geq A$.

Lemma 12 For every $\alpha >0$ there is $A_{1}>0$, $\eta _{1}>0$ and $\varepsilon _{1}>0$ with the following property. If $\lambda \in \overline{\mathbb{Q}}$ is such that $h(\lambda )\leq \eta _{1}$ and $\deg (\lambda)>A_{1} $ then

\[h_{\infty }^{\varepsilon _{1}^{-1}}(\frac{1}{1-\lambda })\leq \alpha\]

Proof: We have

\[h_{\infty }(\frac{1}{1-\lambda })\leq h(\frac{1}{1-\lambda })=h(1-\lambda)\leq h_{f}(\lambda )+h_{\infty }(1-\lambda )\leq h(\lambda )+h_{\infty}(1-\lambda )\]
Hence

\[\frac{1}{\deg (\lambda )}\sum_{x\in \mathcal{O}(\lambda )}\log \frac{1}{|1-x|}=h_{\infty }(\frac{1}{1-\lambda })-h_{\infty }(1-\lambda )\leq h(\lambda )\]
and

\[h_{\infty }^{\varepsilon _{1}^{-1}}(\frac{1}{1-\lambda })=\frac{1}{\deg (\lambda )}\sum_{|1-x|\leq \varepsilon _{1}}\log \frac{1}{|1-x|}\leq h(\lambda )+\frac{1}{\deg (\lambda )}\sum_{|1-x|>\varepsilon _{1}}\log |1-x|\]
Consider the function $f_{\varepsilon _{1}}(z)=\mathbf{1}_{|z-1|>\varepsilon_{1}}\log |1-z|$. It is locally bounded on $\mathbb{C}$. By Bilu's Theorem, for every $\varepsilon _{1}>0$, there must exists $\eta _{1}>0$ and $A_{1}>0$ such that, if $h(\lambda )\leq \eta _{1},$ and $d=\deg (\lambda )>A_{1}$, then
\[\left| \frac{1}{\deg (\lambda )}\sum_{x}f_{\varepsilon_{1}}(x)-\int_{0}^{1}f_{\varepsilon _{1}}(e^{2\pi i\theta })d\theta \right| \leq \frac{\alpha }{3} \]
On the other hand we verify that $\int_{0}^{1}\log |1-e^{2\pi i\theta}|d\theta =0.$ Hence we can choose $\varepsilon _{1}>0$ small enough so that $\left| \int_{0}^{1}f_{\varepsilon _{1}}(e^{2\pi i\theta })d\theta \right| \leq \frac{\alpha }{3}.$ Combining these inequalities with what we obtained before and choosing $\eta _{1}\leq \frac{\alpha }{3},$ we are done.$

Geometric interpretations

We give a geometric interpretation of the minimal norm $E_{v}(F)$.
First let us recall some facts about representations of Chevalley groups. Let $\mathbb{G}$ be a semisimple algebraic group over $\overline{\mathbb{Q}}$. The group $\mathbb{G}$ is a Chevalley group and comes with an associated $\mathbb{Z}$ structure.

We let $\mathfrak{g}_{\mathbb{Z}}$ be a Chevalley order corresponding to $\mathbb{G}$ on the Lie algebra $\mathfrak{g}$ of $\mathbb{G}$ and $\mathfrak{a}$ the associated Cartan subalgebra in $\mathfrak{g.}$ Also let $(Y_{1},...,Y_{d})$ be a Chevalley basis of $\mathfrak{g}_{\mathbb{Z}}$ so that the $Y_{i}$'s for $i\in \lbrack |\Phi ^{+}|+1,|\Phi ^{+}|+r]$ span the admissible lattice $\mathfrak{g}_{\mathbb{Z}}\cap \mathfrak{a}$ of $\mathfrak{a}$ (here $\Phi ^{+}$ is the set of positive roots and $r$ the absolute rank of $\mathbb{G}$). We denote by $T$ the maximal split torus of $\mathbb{G}$ corresponding to $\mathfrak{a}$ and by $\tau $ the Cartan involution.

Given a local field $k,$ we define the Killing norm $||\cdot ||_{Kill,k}$ on $\mathfrak{g}_{k}$ to be the one given by the Killing form $B_{\mathfrak{g}}$ when $k$ is archimedean (i.e. $||X||_{Kill,k}=-B_{\mathfrak{g}}(X^{\tau },X)$) and the one arising from the lattice $\mathfrak{g}_{\mathbb{Z}}\otimes \mathcal{O}_{k}=\mathfrak{g}_{\mathcal{O}_{k}}$ when $k$ is ultrametric (i.e. $||X||_{Kill,k}= \max_{i}|x_{i}|_{k}$ if $X=\sum x_{i}Y_{i}$). This allows us to define what we will call the \textquotedblleft Killing height\textquotedblright\ $h_{Kill}(F)$ for $F\subseteq \mathbb{G}(\overline{\mathbb{Q}})$ by the usual formula (summing over logarithms of matrix norms) where we use the Killing norm at each place.

We denote by $K_{0}$ the stabilizer of $||\cdot ||_{Kill,k}.$ It is a maximal compact subgroup of $\mathbb{G}(k).$ It is also a \textit{good} maximal compact subgroup \ in the sense of \cite[3.3]{BT}, that is $K_{0}$ contains a copy of the Weyl group, so that $N_{K_{0}}(T(k))T(k)=N_{\mathbb{G}(k)}(T(k)).$

Let $V,\rho _{V}$ be a finite dimensional linear representation of $\mathbb{G}$ which is non trivial on each factor of $\mathbb{G}$. By a result of Steinberg there exists an integer lattice, say $V_{\mathbb{Z}}$, of $V$ which is invariant under $\mathbb{G}(\mathbb{Z})$ and which is spanned by a basis $(Y_{1},...,Y_{D})$ made of weight vectors for the action of $T$. When $k$ is ultrametric $V_{\mathcal{O}_{k}}=V_{\mathbb{Z}}\otimes \mathcal{O}_{k}$ defines the following norm on $V_{k}=V_{\mathbb{Z}}\otimes k.$ We denote it by $||X||_{\rho _{V},k}:=\max_{i}|x_{i}|_{k}$ if $X=\sum_{i=1}^{D}x_{i}Y_{i}\in V_{k}.$ When $k$ is Archimedean, then there exists a hermitian scalar product on $V_{k}$ which is invariant under $K_{0}$
and for which $\mathbb{G}(k)$ is stable under taking the adjoint. We denote again by $||\cdot ||_{\rho _{V},k}$ the corresponding hermitian norm. Together these norms define a height function $h_{\rho _{V}}$ on finite subsets of $End(V)$. When $V,\rho _{V}$ is the adjoint representation, the just defined norms and height coincide with the Killing norms and height.

 We set $\mathcal{BT}(\mathbb{G},k)$ to be the Bruhat-Tits building (resp. the symmetric space if $k$ is Archimedean) associated to $\mathbb{G}(k)$. We let $x_{0}$ be the base point of $\mathcal{BT}(SL_{V},k)$ corresponding to the stabilizer of the norm $||\cdot ||_{\rho _{V},k}$. The maximal compact subgroup $K_{0}$ of $\mathbb{G}(k)$ coincides with the the stabilizer of $||\cdot ||_{\rho _{V},k}$ inside $\mathbb{G}(k).$

Let $\ell $ be a finite extension of $k.$ On $\mathcal{BT}(\mathbb{G},\ell )$ we define the distance $d$ to be the standard left invariant distance on $ \mathcal{BT}(\mathbb{G},\ell )$ with the following normalization: if $a\in A, $ then $d(a\cdot x_{0},x_{0})=\sqrt{\sum_{i=1}^{d}(\log |a_{i}|_{k})^{2}}, $ where $\log $ is the logarithm in base $|\pi _{\ell }^{-1}|_{k},$ with $\pi _{\ell }$ a uniformizer for $\mathcal{O}_{\ell }$ when $k$ is non Archimedean, and the standard logarithm if $k$ is Archimedean.
In this normalization, the distance between adjacent vertices on $\mathcal{BT}(\mathbb{G},\ell )$ is of order $1$ and independent of $\ell $ (when $k$ is non Archimedean).

The next proposition shows that the symmetric space or building $\mathcal{BT}(\mathbb{G},k)\simeq \mathbb{G}(k)/K_{0}$ embeds isometrically in $\mathcal{BT}(SL_{V},k)$ as a closed and convex subspace via the orbit map $\mathbb{G}(k)/K_{0}\rightarrow \mathcal{BT}(SL_{V},k),$ $gK_{0}\mapsto g$.

Proposition 13 As above let $k$ be a local field and $\mathbb{G}$ a semisimple $k$-split linear algebraic group, with Cartan decomposition $\mathbb{G}(k)=K_{0}T(k)K_{0}.$ Assume that $\mathbb{G}(k)$ acts properly by isometries on a complete $CAT(0)$ space $X$ in such a way that semisimple elements of $\mathbb{G}(k)$ act by semisimple isometries. Assume that $K_{0}$ fixes a point $p$ in $X$ which belongs to a flat $P$ stabilized by $T(k)$. Then the map $gK_{0}\mapsto g\cdot p$ induces (up to renormalizing the metric on $X$) a $\mathbb{G}(k)$-equivariant isometric embedding $f$ from $\mathcal{BT}(\mathbb{G},k)$ to $X$.

Proof: Let $G=\mathbb{G}(k)$, $T=T(k)$ and $P_{0}$ the $T$-invariant flat in $\mathcal{BT}(\mathbb{G},k)$ containing the base point $p_{0}$ associated to $K_{0}.$ According to the Flat Torus Theorem, there is a unique minimal $T$-invariant flat containing $p$ and its dimension is $\dim T=r=rk(\mathbb{G})$. We may thus assume that $P$ is this minimal flat. However, the normalizer $N_{G}(T)$ permutes the $T$-invariant flats and $N_{G}(T)$ is generated by $T$ and by $N_{G}(T)\cap K_{0}$. It follows that $ N_{G}(T)$ stabilizes $P$. Hence $g\cdot p_{0}\mapsto g\cdot p$ induces an $N_{G}(T)$-equivariant map $f$ between $P_{0}$ and $P$.

Note first that it is enough to show that $f$ is a homothety from $P_{0}$ to $P$. Indeed up to renormalizing the metric in $X,$ we may then assume that $f $ is an isometry from $P_{0}$ to $P,$ i.e. $d(a\cdot p,p)=d(a\cdotp_{0},p_{0}).$
But then for any $g,h\in G$, $d(f(g\cdot p_{0}),f(h\cdot p_{0}))=d(h^{-1}g\cdot p,p)=d(a\cdot p,p)=d(g\cdot p_{0},h\cdot p_{0})$ if $h^{-1}g=k_{1}ak_{2}$ is a Cartan decomposition of $h^{-1}g.$

The fact that $f:P_{0}\rightarrow P$ is a homothety follows from the rigidity of Euclidean Coxeter group actions. Indeed $N_{G}(T)$ contains the affine Weyl group as a co-compact subgroup which acts co-compactly by isometries on both $P_{0}$ and $P.$ But any such action is isometric to the standard Coxeter representation. $\blacksquare$

The relation between the operator norm on $SL(V_{k})$ and the displacement
on $\mathcal{BT}(SL_{V},k)$ in given by the following

Lemma 14 For any $f,g\in SL(V_{k})$ and $x=g^{-1}\cdot x_{0}\in \mathcal{BT}(SL_{V},k),$ letting $\log $ be the logarithm in base $|\pi
_{k}^{-1}|_{k}$, we have
\[\log \left\Vert gfg^{-1}\right\Vert _{\rho _{V},k}\leq d(f\cdot x,x)\leq \sqrt{\dim V}\cdot \log \left\Vert gfg^{-1}\right\Vert _{\rho _{V},k} \]

Proof: Since $d(\cdot ,\cdot )$ is left invariant, we may assume that $g=1$. Then we may write $f=k_{1}ak_{2}$ the Cartan decomposition for $f$. Since the norm is fixed by $K_{0}$ we can assume that $f=a$. Then the estimate is obvious from the normalization we chose for $d(\cdot ,\cdot )$ above. $\blacksquare$

A consequence of this lemma is that the logarithm of the minimal norm of a finite set $F$ is comparable to the minimal displacement of $F$ on $\mathcal{BT}(SL_{V},k).$ We will use a projection argument and the fact that $\mathcal{BT}(SL_{V},k)$ is a $CAT(0)$ space in order to show that the minimal displacement of $F$ is attained on $\mathcal{BT}(\mathbb{G},k).$ More precisely:

Lemma 15: For every finite set $F\in \mathbb{G(}k),$ we have

\[E_{k}(\rho _{V}(F))\leq \inf_{g\in \mathbb{G}(\overline{k})}\left\Vert \rho _{V}(gFg^{-1})\right\Vert _{\rho _{V},k}\leq E_{k}(\rho _{V}(F))^{\sqrt{\dim V}}\]

Corollary 16: Let $d\in \mathbb{N}$ and for a local field $k$ let us denote by $X_{k}$ the symmetric space or Bruhat-Tits building of $PGL_{d}(k).$ We let $d(\cdot ,\cdot )$ be a left invariant Riemannian metric on $X_{\mathbb{C}}$. There is a constant $\varepsilon =\varepsilon (d)>0$ with the following property. Let $K$ be a number field and $F$ a finite subset of $SL_{d}(K)$ which generates a non virtually solvable subgroup $\Gamma $, then either for some finite place $v$ of $K$, the subgroup $\Gamma $ acts (simplicially) without global fixed point on the Bruhat-Tits building $X_{K_{v}},$ or for some embedding $\sigma :K\hookrightarrow \mathbb{C}$

\[\inf_{x\in X_{\mathbb{C}}}\max_{f\in F}d(\sigma (f)\cdot x,x)>\varepsilon.\]

The crucial point here of course is that $\varepsilon $ is independent of
the number field $K$. Thus the Uniform Height Gap Theorem can be seen as a uniform Margulis Lemma for all $S$-arithmetic lattices of a given Lie type. For example, it is uniform over all $SL_{2}(\mathcal{O}_{K})$ where $K$ can vary among all number fields, even though those groups can be lattices of arbitrarily large rank.

Proof: If $F$ fixes a point in the Bruhat-Tits building $X_{k}$ of $SL_{d}$ over a $p$-adic field $k$, then $F$ fixes a vertex of $X_{k}$ (it fixes the vertices of the smallest simplex containing the fixed point). But vertices of $X_{k}$ are permuted transitively by the action of $GL_{d}(k)$. It follows from Lemma 14 that $E_{k}(F)=1.$ Hence if $F$ fixes a point on each $X_{k}$ for $k$ non archimedean, then $e_{f}(F)=0.$ Hence by the Uniform Height Gap theorem we must have $e_{\infty }(F)>\varepsilon .$ Thus there exists an embedding $\sigma $ of $K$ in $\mathbb{C}$ such that $\log E_{\mathbb{C}}(\sigma (F))>\varepsilon .$ Then by Lemma 14, every point of $X_{\mathbb{C}}$ must be moved by at least $\varepsilon $ by some element of $F$.