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)