Speaker: Rostislav Grigorchuk, Distinguished prof. of Mathematics at Texas A&M.

Title: Fractal, liftable and  scale groups.

Abstract:  Scale  groups  are  closed subgroups of the group of isometries of a regular tree that fix an end of the tree and are vertex-transitive.  They  play  an  important  role  in the study  of  locally  compact  totally  disconnected  groups as  was  recently observed  by  P-E.Caprace and  G.Willis.  In  the  past  they  were  studied   in  the  context  of abstract  harmonic  analysis,  random  walks   and  amenability. It  is a  miracle  that  they  are  closely  related  to  fractal    groups,  a  special  subclass  of  self-similar  groups.

In my  talk  I  will discuss  two ways  of  building  scale  groups.  One  is  based  on  the  use  of  scale-invariant  groups studied  by  V.Nekrashevych  and  G.Pete,  and  a second is  based  on  the  use  of  liftable  fractal groups.  The  examples  based on both  approaches  will  be  demonstrated  using  such groups  as  Lamplighter, Basilica,  Hanoi Tower Group, Group  of  Intermediate  Growth (between  polynomial  and  exponential) constructed  by the speaker  in  1980,  and  GGS-groups. Additionally,  the group  of  isometries  of  the  ring  of  integer  p-adics  and group of  dilations  of  the  field  of  p-adics  will  be  mentioned  in  the  relation  with the  discussed  topics.