Universidade de Brasília

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.