The authors study actions and modules of maximal growth over finitely generated free associative algebras, free groups, free monoids and free group algebras. Many interesting results are proved. We will point out some of them. For example the growth function of the free monoid $W_r$ of rank $r$ and free associative algebra $A_r$ of rank $r$ can be written as $α(n)r^n$, where $α(n)$ is a function converging to a limit $C_0$, $C_0\ge 0$, at infinity. The growth is maximal if and only if $C_0>0$. In the case of the free group $F_r$ of rank $r$ and its group algebra $\cal F_r$ the growth function $g(n)$ of a finitely generated action is of the form $α(n)(αr-1)^n$, with $α(n)$ satisfying the same condition as above. The authors show that for the free associative algebra $A_r$ of rank $r$ if $N$ is a submodule of a finitely generated infinite-dimensional $A_r$-module $M$ and $N$ is of finite codimension, then $N$ is finitely generated and the growth of $M$ is the same as the growth of $N$. The modules are considered over a field $F$. The authors prove that in every module $M$ over a free group algebra of rank $r>1$ there is the largest submodule $N$ whose growth is not maximal and the growth of every non-zero submodule of $M/N$ is maximal. Thus if $R$ is a free associative algebra or a free group algebra of rank $r>1$ over a field then the modules none of whose submodules have maximal growth form a radical class, while the modules with every non-zero submodule of maximal growth form a semisimple class. The authors prove that any finitely generated monoid $W_r$, $r>1$ of maximal growth is faithful. Analogously any finitely generated module over the free associative algebra $A_r$ of rank $r$ of maximal growth is faithful. The authors show the following interesting result. Let $R$ be a free associative algebra over a field $F$ of rank greater than $1$ and let $M$ be a graded $R$-module of maximal growth. Then $M$ has a graded factor module of maximal growth which in addition is a nil module. In other sections the authors also prove some results about residually finite modules of maximal growth, graphs of actions. Many properties of graphs of actions are proved. Topological approach to maximal growth is discussed.
Reviewer: Vitaly Linchenko (Yerakhtur)