@AndreiJaikin I wish he can put his answer himself here! This answers (Problem 17.17 of THE KOUROVKA NOTEBOOK, No. 18; arxiv.org/pdf/1401.0300v8.pdf) proposed by G.M.Bergman: If a finitely generated group $G$ has $n < \infty$ maximal subgroups, must $G$ be finite? In particular, what if $n = 3$?

@AndreiJaikin. Andrei Jaikin-Zapirain has sent an answer in Group-Pub-forum at 2016/Aug/21 as follows: An example of an infinite group with 3 maximal subgroups can be extracted from the paper Ershov, Mikhail; Jaikin-Zapirain, Andrei Groups of positive weighted deficiency and their applications. J. Reine Angew. Math. 677 (2013), 71–134. A 2-generated 2-LERF group (see Section 7 of the paper) has this property because its maximal subgroups have index 2.

@RobertChamberlain The term "Baer radical" is also used for the set of elements $x$ of a group $G$ such that the cyclic subgroup $<x>$ is a subnormal subgroup in $G$: See page 61 of [Finiteness Conditions and Generalized Soluble Groups, Part 1 By Derek Robinson, Springer, 1970] books.google.com/…

@YCor Another way, if we assume that the size of the Folner set (which is depending to $A$ and $\epsilon$) is bounded above by a function $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ of $\epsilon$ only. Under what conditions on $f$ one can conclude that the group is virtually solvable.

@EhudMeir : Yes I mean that we have 3 edges in total, both inbound and outbound.

@EhudMeir : Why there is an arc from $g$ to $x^{-1}g$?

@EhudMeir : The graph is directed.

@verret : Two vertices $u$ and $v$ "are adjacent" if there is an arc from $u$ to $v$ or there is an arc from $v$ to $u$.