## Notes of Pierre-Emmanuel Caprace’s first lecture 05-04-2017

Exotic lattices and simple locally compact groups

1. BMW complexes and groups

I like locally compact groups when they contain lattices.

1.1. Two examples of lattices

Let

$\displaystyle \begin{array}{rcl} \Gamma_1&=&\langle a,b,x,y\,|\,axay,\,ax^{-1}bx^{-1},\,ay^{-1}b^{-1}y^{-1},\,bxby^{-1}\rangle \\ \Gamma_2&=&\langle a,b,x,y\,|\,axay,\,ax^{-1}by^{-1},\,ay^{-1}b^{-1}x^{-1},\,bxb^{-1}y^{-1}\rangle \end{array}$

Although similar looking, these groups are pretty different.

${\Gamma_1}$ was studied by Stix-Vdovina. They showed that ${\Gamma_1}$ is hereditarily just infinite (infinite and every proper quotient is finite, and this property is inherited by all finite index subgroups). It is not virtually abelian. Hence ${\Gamma_1}$ (and all its finite index subgroups) has finite abelianization. It is linear over a field of characteristic 3. In particular, it is residually finite.

${\Gamma_2}$ maps onto the dihedral group ${D_\infty}$ by ${a\mapsto s}$, ${b\mapsto s}$, ${x\mapsto sts}$, ${y\mapsto t}$. Hence ${\Gamma_2}$ has infinite abelianization. It is not residually finite (we shall spend time proving this).

There exist just infinite but not hereditarily just infinite groups (e.g. wreath products ${S\wr C_2}$ where ${S}$ is infinite and simple).

${\Gamma_1}$ and ${\Gamma_2}$ are both subgroups of ${Aut(T_4)\times Aut(T_4)}$. They act simply transitively on vertices of the square complex ${T_4\times T_4}$. Each quotient ${Y_i=\Gamma_i \setminus X}$ is a square complexe with 1 vertex, 4 edges and 4 squares.

Definition 1 A BMW complex is a square complex with one vertex, ${m+n}$ edges, ${mn}$ squares, and such that the link of the vertex is the complete bipartite graph ${K_{2m,2n}}$.

A BMW is is the fundamental group of a BMW complex. The pair ${(m,n)}$ is called the order.

A complete square complex (CSC) is a square complex all of whose links are complete bipartite graphs.

BMW stands for Burger, Mozes and Wise.

1.2. Characterisation of BMW complexes

Fact. If ${Y}$ is a CSC, its universal cover is a product of two trees. If ${Y}$ is furthermore a BMW-complex of order ${(m,n)}$, then the universal cover is a product ${T_{2m}\times T_{2n}}$.

Lemma 2 ${\Gamma}$ is a BMW group iff ${\Gamma}$ is torsion free and ${\Gamma}$ admits a Cayley graph isomorphic to the 1-skeleton of the product of two trees.

Examples

1. Products of free groups are BMW.
2. BMW groups of order ${(1,1)}$ are the fundamental groups of the 2-torus and of the Klein bottle.

Definition 3 A BMW complex is reducible if some finite cover is a product of two graphs.

Fact. Every BMW complex of order ${(1,n)}$ is reducible.

Examples ${\Gamma_1}$ and ${\Gamma_2}$ are irreducible, of order ${(2,2)}$.

2. History

In 1992, Shahar Mozes produced, for all distinct primes ${p}$ and ${\ell}$ congruent to 1 mod 4, an irreducible BMW group of order${(\frac{p+1}{2},\frac{\ell+1}{2}}$. This group is a lattice in ${\times PGl(2,{\mathbb Q}_p)\times PGl(2,{\mathbb Q}_p)}$. It is hereditarily just infinite by Margulis’ normal subgroup theorem.

In 1994, Meenaxi Bhattacharjee constructed a finitely presented infinite group without finite quotients. It is an amalgamation of two free groups ${F_3}$ over an ${F_{13}}$ that acts on ${T_6}$. For this, she arranged the embeddings of ${F_{13}}$ so that for some element ${a}$, ${a^2}$ is conjugate to ${a^5}$. This kills any finite quotient.

In 1996, Wise showed that there exists a non-residually finite BMW group of order ${(3,4)}$.

Using Gromov’s hyperbolization, B. Hu showed that there exist closed manifolds with a locally ${CAT(0)}$ metric and a non-residually finite fundamental group. Question: what about NPC manifolds?

In 1997, Burger and Mozes proved that for all ${m\geq 109}$ and all ${n\geq 150}$, there exsts a virtually BMW group of order ${(m,n)}$.

In 2004, Rattaggi generalized Mozes’ arithmetic construction to all pairs of distinct odd primes. He improved Burger and Mozes’ bound to ${5}$ (for ths, he relied on Wise’s examples.

In 2004, motivated by operator algebras, Kimberley-Robertson classified BMW-complexes of order ${(2,2)}$. They found 52 of them. This led to a classification of

Comparing with Rattaggi’s work, I could show that at least 50 of the 52 are reducible. The relaining ones are ${\Gamma_1}$ and ${\Gamma_2}$.

In 2013, Stix and Vdovina showed that for every prime power, there exists an irreducible BMW groups of order ${(\frac{q+1}{2},\frac{q+1}{2})}$, it is a lattice in ${PGl(2,\mathbb{F}_q((t))) \times PGl(2,\mathbb{F}_q((t)))}$.

Recently, Nicolas Radu showed that ${\Gamma_2}$ is irreducble and not residually finite. This leads to virtually simple examples starting from order ${(3,4)}$.

3. Counting commensurability classes of groups

I see an analogy with closed real hyperbolic manifolds.

3.1. Counting hyperbolic manifolds

Let ${n\geq 4}$. Given ${v>0}$, let ${C_n^c(v)}$ be the number of commensurability classes of closed real hyperbolic ${n}$-manifolds having a representative of volume ${\leq v}$.

Theorem 4 (Gelander-Levitt 2014) This number is super-exponential,

$\displaystyle \begin{array}{rcl} v^{av}\leq C_n^c(v)\leq v^{bv}. \end{array}$

On the other hand, the number of arithmetic commensurability classes is conjectured to be polynomial. The best known result is a bit weaker.

Theorem 5 (Belolipetsky 2007)

$\displaystyle \begin{array}{rcl} C_m^{\mathrm{arithmetic}}(v)\leq v^{\beta(\log v)^\epsilon}. \end{array}$

3.2. Counting BMW groups

Conjecture. Let ${C_n^c(m)}$ be the number of commensurability classes of BMW groups of order ${(m,m)}$. Then ${C_n^c(m)}$ is super-exponential. The number of arithmetic ones is polynomial.

Stix-Vdovina provide a linear lower bound for the arithmetic ones.

The goal of these lectures is to substantiate this conjecture.