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
Although similar looking, these groups are pretty different.
was studied by Stix-Vdovina. They showed that 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 (and all its finite index subgroups) has finite abelianization. It is linear over a field of characteristic 3. In particular, it is residually finite.
maps onto the dihedral group by , , , . Hence 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 where is infinite and simple).
and are both subgroups of . They act simply transitively on vertices of the square complex . Each quotient is a square complexe with 1 vertex, 4 edges and 4 squares.
Definition 1 A BMW complex is a square complex with one vertex, edges, squares, and such that the link of the vertex is the complete bipartite graph .
A BMW is is the fundamental group of a BMW complex. The pair 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 is a CSC, its universal cover is a product of two trees. If is furthermore a BMW-complex of order , then the universal cover is a product .
Lemma 2 is a BMW group iff is torsion free and admits a Cayley graph isomorphic to the 1-skeleton of the product of two trees.
- Products of free groups are BMW.
- BMW groups of order 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 is reducible.
Examples and are irreducible, of order .
In 1992, Shahar Mozes produced, for all distinct primes and congruent to 1 mod 4, an irreducible BMW group of order. This group is a lattice in . 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 over an that acts on . For this, she arranged the embeddings of so that for some element , is conjugate to . This kills any finite quotient.
In 1996, Wise showed that there exists a non-residually finite BMW group of order .
Using Gromov’s hyperbolization, B. Hu showed that there exist closed manifolds with a locally metric and a non-residually finite fundamental group. Question: what about NPC manifolds?
In 1997, Burger and Mozes proved that for all and all , there exsts a virtually BMW group of order .
In 2004, Rattaggi generalized Mozes’ arithmetic construction to all pairs of distinct odd primes. He improved Burger and Mozes’ bound to (for ths, he relied on Wise’s examples.
In 2004, motivated by operator algebras, Kimberley-Robertson classified BMW-complexes of order . 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 and .
In 2013, Stix and Vdovina showed that for every prime power, there exists an irreducible BMW groups of order , it is a lattice in .
Recently, Nicolas Radu showed that is irreducble and not residually finite. This leads to virtually simple examples starting from order .
3. Counting commensurability classes of groups
I see an analogy with closed real hyperbolic manifolds.
3.1. Counting hyperbolic manifolds
Let . Given , let be the number of commensurability classes of closed real hyperbolic -manifolds having a representative of volume .
Theorem 4 (Gelander-Levitt 2014) This number is super-exponential,
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)
3.2. Counting BMW groups
Conjecture. Let be the number of commensurability classes of BMW groups of order . Then 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.