---
tag: 6G73DJH9T6
kind: statement
role: theorem
name: Preissmann's theorem
msc: [53C20, 53C22]
uses: [7CBT4SM83Y, 1X42JAA349, S1XK3HXHFX, 428R02EYWS, NSYMQMHPDN]
about: [1X42JAA349, S1XK3HXHFX]
sources:
  - url: https://en.wikipedia.org/wiki/Preissmann%27s_theorem
same_as:
  wikidata: Q7240001
  wikipedia: Preissmann's theorem
---

## Statement

Let $(M, g)$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] with negative [[sectional curvature|1X42JAA349]].
Then every nontrivial abelian subgroup of the [[fundamental group|S1XK3HXHFX]] $\pi_1(M)$ is infinite cyclic.

## Remarks

The idea of the proof: by the [[Cartan–Hadamard theorem|428R02EYWS]], the universal cover $\tilde M$ is diffeomorphic to $\R^n$ and $\pi_1(M)$ acts on it by deck isometries.
Each nontrivial element has infinite order and, by compactness, moves points along a unique invariant [[geodesic|NSYMQMHPDN]], its axis.
Negative curvature forces two commuting elements to have the same axis.
So an abelian subgroup acts by translations on one geodesic line, and a discrete group of translations of a line is infinite cyclic.

Example: let $M_1$ and $M_2$ be compact manifolds whose fundamental groups contain elements of infinite order, such as $M_1 = M_2 = S^1$.
Then $\pi_1(M_1 \times M_2)$ contains $\Z^2$, so $M_1 \times M_2$ carries no metric of negative sectional curvature.

Negative curvature is needed: the flat torus $\R^2 / \Z^2$ has curvature $0$ and fundamental group $\Z^2$.

## Explanations

### L1 human

In a negatively curved compact space, two loops that commute must go "in the same direction": both are powers of a single loop.
So the fundamental group of such a space contains no copy of $\Z^2$, the group of loops on a torus.

### L2 human

Preissmann's theorem is a group-theoretic obstruction to negative curvature.
It rules out, for example, every product of two compact manifolds whose fundamental groups have elements of infinite order.
It is an early instance of the link between negative curvature and the algebra of the fundamental group that later led to Gromov's hyperbolic groups.