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).

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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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