480 research outputs found
Branch groups with infinite rigid kernel
A theoretical framework is established for explicitly calculating rigid
kernels of self-similar regular branch groups. This is applied to a new
infinite family of branch groups in order to provide the first examples of
self-similar, branch groups with infinite rigid kernel. The groups are analogs
of the Hanoi Towers group on 3 pegs, based on the standard actions of finite
dihedral groups on regular polygons with odd numbers of vertices, and the rigid
kernel is an infinite Cartesian power of the cyclic group of order 2, except
for the original Hanoi group. The proofs rely on a symbolic-dynamical approach,
related to finitely constrained groups.Comment: Comments welcome
Asymptotic aspects of Schreier graphs and Hanoi Towers groups
We present relations between growth, growth of diameters and the rate of
vanishing of the spectral gap in Schreier graphs of automaton groups. In
particular, we introduce a series of examples, called Hanoi Towers groups since
they model the well known Hanoi Towers Problem, that illustrate some of the
possible types of behavior.Comment: 5 page
Diameters, distortion and eigenvalues
We study the relation between the diameter, the first positive eigenvalue of
the discrete -Laplacian and the -distortion of a finite graph. We
prove an inequality relating these three quantities and apply it to families of
Cayley and Schreier graphs. We also show that the -distortion of Pascal
graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain
estimates for the convergence to zero of the spectral gap as an application of
the main result.Comment: Final version, to appear in the European Journal of Combinatoric
Finite self-similar p-groups with abelian first level stabilizers
We determine all finite p-groups that admit a faithful, self-similar action
on the p-ary rooted tree such that the first level stabilizer is abelian. A
group is in this class if and only if it is a split extension of an elementary
abelian p-group by a cyclic group of order p.
The proof is based on use of virtual endomorphisms. In this context the
result says that if G is a finite p-group with abelian subgroup H of index p,
then there exists a virtual endomorphism of G with trivial core and domain H if
and only if G is a split extension of H and H is an elementary abelian p-group.Comment: one direction of theorem 2 extended to regular p-group
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