Virtual Endomorphisms of Groups

29 April 2014
A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$ is a subgroup of $G$ of fi…nite index $m$: A recursive construction using $f$ produces a so called state-closed (or, self-similar in dynamical terms) representation of $G$ on a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$; i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant. Examples of state-closed groups are the Grigorchuk 2-group and the Gupta- Sidki $p$-groups in their natural representations on rooted trees. The affine group $Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed representations. Yet another example is the free nilpotent group $G = F (c; d)$ of class c, freely generated by $x_i (1\leq  i \leq  d)$: let $H = \langle x_i^n | \ (1 \leq  i \leq  d) \rangle$ where $n$ is a fi…xed integer greater than 1 and $f$ the extension of the map $x^n_i \rightarrow x_i$ $(1 \leq  i \leq  d)$. We will discuss state-closed representations of general abelian groups and of …nitely generated torsion-free nilpotent groups.