Actions of maximal growth

Abstract

We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong "infiniteness" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional.

Filtrations and distortion in infinite-dimensional algebras

“…If B is a subalgebra in a finitely generated algebra A, and β ∼ α ∩ B where α is a degree filtration of A then, as noted in Introduction just after Definition 2, β is tame. This proves the "if" parts in claims (1) and (2).…”

“…Now since constant C in our present proof is the same as in the proof of Theorem 7, we can easily pass from present claim (1) to present claim (2) in the same manner as we did in the case of associative algebras. 2…”

“…,n, in the previous equation contains a leading word of a relation in S ∪ R, as a subword. Proving by contradiction, let us assume, say, that u(1) f (2)v(1) contains such leading word w. If w has no letters x, y then w is a subword in f (2), which is impossible. Otherwise, w = u f i v, for some i, where the "wings" u and v are in the set of non-overlapping words M. As just above, we cannot have w completely inside f (2) and so either u has overlapping with u(1) (hence, equal to u(1)) or v has overlapping v(1) (hence, equal to u(1)), or both.…”

“…(2) If the answer to the previous question is "yes", can one choose A finitely presented? If not, indicate conditions ensuring that the answer is still "yes".…”

“…To prove the "only if" part in claim (2), notice that the previous proof works for any number of elements in the set X , #X 2. But if #X is big enough, as specified in the first paragraph of this subsection, then the growth of the set M can be made faster than any exponential function.…”