\titleofpresentation{On Brian Hartley problem on maximal subgroups in branch groups.}

\presenter{Rostislav Grigorchuk}

I will speak about the problem posted by Brian Hartley on existence of maximal subgroups
in certain branch self-similar groups.

The solution of  the  problem was  found  by  Ekaterina  Pervova  for the main case,  that   interested Bryan Hartley (including the first example of a group of intermediate growth,
and Gupta-Sidki $p$-groups). The  topic  was  farther developed
in  2002  in joint  work  of  J.S.Wilson  and    speaker.  Namely  the  result  of  Pervova was
generalized to  all  groups  which are  abstractly  commensurable  with  the $3$-generated
$2$-group $\mathcal G$  of  intermediate growth.  Unexpected consequence  of  this (and  the  developed  technique)
is  the  subgroup  separability  of   the  group  $\mathcal G$   (quite  rare  property in group  theory)  and
the  fact  that  any  infinite finitely  generated  subgroup of $\mathcal G$  is abstractly
commensurable  to  $\mathcal G$. The  overview  of  all  this  results, some  new observation  and  a list  of  open  questions  will  
be  presented in  the  talk.

\presenteraddress{Texas A\&M University}

\email{grigorch@math.tamu.edu}

\web{www.math.tamu.edu/~grigorch}


