---
tag: SP421Y498C
kind: statement
role: theorem
name: Wang's finiteness theorem
msc: ["22E40"]
uses: [BTYF5Y5PFY, XJS07AJ50Y, VY0HXD3GC4, C34RZMKKA4, V2HTMG4KT1]
about: [BTYF5Y5PFY]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)
    locator: Rigidity results
---

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and without compact factors.
Assume that no simple factor of $G$ is locally isomorphic to $\mathrm{SL}_2(\R)$ or to $\mathrm{SL}_2(\C)$.
Fix a [[Haar measure|VY0HXD3GC4]] on $G$, and for a [[lattice|BTYF5Y5PFY]] $\Gamma$ of $G$ let $\operatorname{covol}(\Gamma)$ be the measure of $G/\Gamma$ for the induced invariant measure.
Then for every $v > 0$ there are only finitely many lattices $\Gamma$ in $G$ with $\operatorname{covol}(\Gamma) \le v$, up to conjugation by elements of $G$.

## Remarks

The result is due to H. C. Wang.
The proof combines two facts.
The [[Kazhdan–Margulis theorem|C34RZMKKA4]] gives a fixed neighbourhood of the identity that every lattice avoids up to conjugation, which makes the set of lattices of bounded covolume compact up to conjugation.
[[Weil's local rigidity theorem|V2HTMG4KT1]] and its extension to non-uniform lattices make each lattice isolated up to conjugation, and a compact set of isolated points is finite.
The excluded factors are necessary.
Fix a genus $g \ge 2$.
In $\mathrm{PSL}_2(\R)$ the fundamental groups of closed hyperbolic surfaces of genus $g$ all have covolume $4\pi(g - 1)$, by the Gauss–Bonnet theorem.
Yet they form uncountably many conjugacy classes, parametrised by [[Teichmüller space|ZZGC6BZNJ5]].
In $\mathrm{PSL}_2(\C)$ hyperbolic Dehn filling gives infinitely many closed hyperbolic $3$-manifolds whose volumes are bounded by the volume of one cusped hyperbolic $3$-manifold.

## Explanations

### L1 human

A lattice gives a finite-volume space $G/\Gamma$, and one can measure its size.
The theorem says that, for most semisimple groups, only finitely many essentially different lattices have size below any given bound.
So lattices are sparse, unlike in the plane, where lattices can be deformed continuously.
For example, there are only finitely many closed hyperbolic manifolds of dimension at least $4$ with volume below any fixed number.

### L2 human

Wang's theorem is the global consequence of two local facts: lattices cannot contain elements too close to the identity in a uniform way, and lattices cannot be deformed.
It fails for factors of type $\mathrm{SL}_2(\R)$, where lattices deform continuously.
It also fails for $\mathrm{SL}_2(\C)$, where hyperbolic Dehn filling produces infinitely many lattices of bounded covolume.
It is a starting point for counting lattices of bounded covolume, and for the study of the smallest covolume of a lattice in a given group.