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.

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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 http://www.math.ens.fr/metric2011/
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