# Modulus of an automorphism

A positive real number associated to an automorphism of a locally compact group. Let $G$ be such a group and let $\alpha$ be an automorphism of $G$, regarded as a topological group. Then the modulus of $\alpha$ is defined by

$$\mathop{\rm mod} _ {G} ( \alpha ) = \ \frac{\mu ( \alpha S ) }{\mu ( S) } ,$$

where $\mu$ is left-invariant Haar measure on $G$ and $S$ is any compact subset of $G$ with positive measure (indeed, $\mathop{\rm mod} _ {G} ( \alpha )$ does not depend on $S$). If $G$ is compact or discrete, then $\mathop{\rm mod} _ {G} ( \alpha ) \equiv 1$, since for a compact group one can put $S = G$, and for a discrete group one can take $S = \{ 1 \}$, where $1$ is the identity element of $G$.

If $\alpha$ and $\beta$ are two automorphism of $G$, then

$$\mathop{\rm mod} _ {G} ( \alpha \cdot \beta ) = \ \mathop{\rm mod} _ {G} ( \alpha ) \ \mathop{\rm mod} _ {G} ( \beta ) .$$

If $\Gamma$ is a topological group which acts continuously on $G$ by automorphisms, then the associated homomorphism $\pi : \Gamma \rightarrow \mathop{\rm Aut} G$ defines a continuous homomorphism $\mathop{\rm mod} _ {G} \circ \pi : \Gamma \rightarrow \mathbf R _ {+} ^ {*}$, where $\mathbf R _ {+} ^ {*}$ is the multiplicative group of positive real numbers. In particular, if $\Gamma = G$ and $\pi ( g) ( x) = g x g ^ {-} 1$, then $\pi \circ \mathop{\rm mod} _ {G} : G \rightarrow \mathbf R _ {+} ^ {*}$ is a continuous homomorphism. This homomorphism is trivial if and only if the left-invariant Haar measure on $G$ is simultaneously right invariant. Groups satisfying the latter condition are called unimodular.

If $K$ is a locally compact skew-field, then each non-zero element $a \in K$ defines an automorphism $\mu ( a)$ of the additive group of $K$ via multiplication by $a$. The function $\mathop{\rm mod} _ {K} \circ \mu : K \setminus \{ 0 \} \rightarrow \mathbf R _ {+} ^ {*}$ is used in the study of the structure of locally compact skew-fields.

#### References

 [1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) [2] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940) [3] A. Weil, "Basic number theory" , Springer (1974)
How to Cite This Entry:
Modulus of an automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_automorphism&oldid=47878
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article