The main topic of this thesis is the isoperimetric profile of algebras, introduced by Gromov in [21]. This is an asymptotic geometric invariant of algebras which has a strong connection with the property of amenability. We pursue a systematic study of this invariant, which will lead us to new insights into the phenomenon of amenability of algebras. Also, we show the connections of the isoperimetric profile with other invariant of algebras, like the GK-dimension, the growth and the lower transcendence degree. We will use our tools to provide answers to some questions on these topics.
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