## Notes of Nikolay Nikolov’s Cambridge lecture 10-05-2017

Homology torsion growth of higher rank lattices

Let ${\Gamma}$ be a residually finite group. Let ${\Gamma_i}$ be a tower of exhausting, not necessarily normal, finite index subgroups. Say this is a Farber sequence if the permutation actions of ${\Gamma}$ on ${\Gamma/\Gamma_i}$ form a sofic approximation of ${\Gamma}$. Equivalently, if ${\Gamma}$ acts essentially freely on the profinite space ${\lim \Gamma/\Gamma_i}$.

Let ${G}$ be a center free simple Lie group. Let ${\Lambda_i}$ be a sequence of lattices, ${M_i=\Lambda_i \setminus X}$ the corresponding locally symmetric orbifold.

Definition 1 The sequence ${\Lambda_i}$ is Farber (or Benjamini-Schramm) if ${\forall R>0}$, the ${R}$-thin part of ${M_i}$ has relative volume tending to 0.

Theorem 2 (7 authors) If ${G}$ has real rank ${\geq 2}$, every sequence of lattices of increasing covolumes is Farber.

This relies on the Stuck-Zimmer theorem.

1. Growth

I am interested in the growth of numerical invariants of such lattices.

1.1. Betti numbers

For type ${F_{n+1}}$ groups, normalized Betti numbers

$\displaystyle \frac{dim_{\mathbb Q} H_n(\Gamma_i,{\mathbb Q})}{[\Gamma:\Gamma_i]}$

converge (Luck, Farber).

Theorem 3 For uniformly discrete lattices (ie injectivity radius bounded below), with covolume tending to infinity, normalized Betti numbers converge.

1.2. Rank

Let ${d(\Gamma)}$ be the minimum size of a generating set. It can grow at most linearly, so we define the rank gradient

$\displaystyle \begin{array}{rcl} RG(\Gamma,(\Gamma_i)):=\lim \frac{d(\Gamma_i)-1}{[\Gamma:\Gamma_i]} \end{array}$

The limit exists, but may depend on the sequence.

Definition 4 Say ${\Gamma}$ is right-angled if admits a generating set whose elements ${g_i}$ have infinite order and commute sequentially: ${[g_i,g_{i+1}]=1}$.

Theorem 5 (Gaboriau) If ${\Gamma}$ is right angled, then ${\Gamma}$ has fixed price 1, ie every ergodic free action has cost 1.

Theorem 6 (Abert-Nikolov) If ${\Gamma_i}$ is a Farber chain, then ${RG(\Gamma,(\Gamma_i))=cost(\Omega)-1}$, where ${\Omega}$ is the profinite space ${\lim \Gamma/\Gamma_i}$.

Hence Farber rank gradient vanishes for right-angled groups.

1.3. Torsion in ${H_1}$

If ${\Gamma}$ is finitely presented, the size ${T(\Gamma_i)}$ of torsion in ${H_1(\Gamma,{\mathbb Z})}$ grows at most exponentially. However, it is not clear wether

$\displaystyle \begin{array}{rcl} \frac{\log T(\Gamma_i)}{[\Gamma:\Gamma_i]} \end{array}$

converges.

Torsion has been studied for a while (Lackenby,…). Lower bound are hard.

Theorem 7 (Abert-Gelander-Nikolov) Let ${\Gamma}$ be right-angled and ${\{\Gamma_i\}}$ be a Farber sequence. Then

$\displaystyle \begin{array}{rcl} RG(\Gamma,(\Gamma_i))=\lim \frac{\log T(\Gamma_i)}{[\Gamma:\Gamma_i]}=0. \end{array}$

This borrows ideas from Gaboriau. It is graph theoretic: removing edges does not change the metric too much…

Theorem 8 (Abert-Gelander-Nikolov) ${G=Sl(n,{\mathbb R})}$ and ${SO(n,2)}$ have right-angled cocompact lattices.

2. A bold conjecture

This suggests the following

Conjecture. Let ${\{\Lambda_i\}}$ be a sequence of lattices in a higher rank simple Lie group, with covolumes tending to infinity. Then rank is sublinear and torsion is subexponential.

Mark Shusterman has made the following observation. Since ${H_1}$ is finite (property (T)), its size behaves rather easily under coverings. Sharma and Venkataramana showed that, for non uniform lattices, ${d(\Lambda)\leq d(\hat \Lambda)+3}$. ${\hat\Lambda}$ is understood thanks to CSP. Combined with Belolipetsky-Lubotzky’s work on lattice growth, this proves the conjecture for non-uniform lattices.

2.1. Rank 1

Bergeron-Venkatesh conjecture that for principal congruence subgroups in ${PSl(2,{\mathbb C})}$, ${\lim \frac{\log T(\Gamma_i)}{[\Gamma:\Gamma_i]}}$ is nonzero.

Bader-Gelander-Sauer give an exponential upper bound for ${SO(n,1)}$, ${n\geq 4}$. They show that in ${SO(3,1)}$, there is no upper bound on torsion in terms of covolume. On the other hand, there is an exponential bound for 3d arithmetic lattices (Fraczyk).