Coloquio Cuernavaca, noviembre de 2023

Abstract:  Scale  groups  are  closed subgroups of the group of isometries of the 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  observed  by  G.Willis.   It  is a  miracle  that  they  are  closely  related  to  self-replicating  groups,  a  special  subcalls  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  groups  --  a  special  class  of  self-replicating  groups.  The  examples  based on both  approaches  will  be  demonstrated  including  the  Lamplighter   group  L  and a torsion  2-group  of  intermediate  growth G.  It  will  be  shown  that its  finitely  presented non elementary  amenable  relative G'   gives  the example  of a  scale  group acting  2-transitively   on a  punctured  boundary.

Additionally  the group  of  isometries  of  the  ring  of  integer  p-adics  and group of  dilations  of  the  field  of  p-adic  will  be  mentioned  in  the  relation  with the  discussed  topics.