Homology torsion growth of higher rank lattices
Let be a residually finite group. Let be a tower of exhausting, not necessarily normal, finite index subgroups. Say this is a Farber sequence if the permutation actions of on form a sofic approximation of . Equivalently, if acts essentially freely on the profinite space .
Let be a center free simple Lie group. Let be a sequence of lattices, the corresponding locally symmetric orbifold.
Definition 1 The sequence is Farber (or Benjamini-Schramm) if , the -thin part of has relative volume tending to 0.
Theorem 2 (7 authors) If has real rank , every sequence of lattices of increasing covolumes is Farber.
This relies on the Stuck-Zimmer theorem.
I am interested in the growth of numerical invariants of such lattices.
1.1. Betti numbers
For type groups, normalized Betti numbers
converge (Luck, Farber).
Theorem 3 For uniformly discrete lattices (ie injectivity radius bounded below), with covolume tending to infinity, normalized Betti numbers converge.
Let be the minimum size of a generating set. It can grow at most linearly, so we define the rank gradient
The limit exists, but may depend on the sequence.
Definition 4 Say is right-angled if admits a generating set whose elements have infinite order and commute sequentially: .
Theorem 5 (Gaboriau) If is right angled, then has fixed price 1, ie every ergodic free action has cost 1.
Theorem 6 (Abert-Nikolov) If is a Farber chain, then , where is the profinite space .
Hence Farber rank gradient vanishes for right-angled groups.
1.3. Torsion in
If is finitely presented, the size of torsion in grows at most exponentially. However, it is not clear wether
Torsion has been studied for a while (Lackenby,…). Lower bound are hard.
Theorem 7 (Abert-Gelander-Nikolov) Let be right-angled and be a Farber sequence. Then
This borrows ideas from Gaboriau. It is graph theoretic: removing edges does not change the metric too much…
Theorem 8 (Abert-Gelander-Nikolov) and have right-angled cocompact lattices.
2. A bold conjecture
This suggests the following
Conjecture. Let 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 is finite (property (T)), its size behaves rather easily under coverings. Sharma and Venkataramana showed that, for non uniform lattices, . 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 , is nonzero.
Bader-Gelander-Sauer give an exponential upper bound for , . They show that in , 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).