2
$\begingroup$

Let $T$ denote the unique (countably infinite) tree in which every vertex has degree $3$ and let $G=Aut(T)$ be the group of graph automorphisms of $T$.

I'm interested in the properties of this group, especially in the normal subgroups of $G$. Is there a known classification of all normal subgroups? If not, what is known and what is conjectured about them?

$\endgroup$
3
  • $\begingroup$ Is your tree planar or abstract? $\endgroup$ Feb 21, 2014 at 16:55
  • $\begingroup$ Abstract. In particular $G$ is the full automorphism group of $T$. $\endgroup$
    – Dominik
    Feb 21, 2014 at 16:57
  • $\begingroup$ You might want to look up branch groups; see for example the book by Bartholdi, Grigorchuk and Sunik arxiv.org/pdf/math/0510294.pdf. $\endgroup$ Feb 21, 2014 at 19:21

2 Answers 2

11
$\begingroup$

Let $G_+$ be the index 2 subgroup of $G$, consisting of automorphisms which act on $T$ without inversions, which means that if such an automorphism preserves an edge, it fixes this edge pointwise. Equivalently, this is the subgroup of $G$ generated by (pointwise) edge-stabilizers. It follows from a paper by J.Tits in 1970 ("Sur le groupe des automorphismes d'un arbre") that the subgroup $G_+$ is simple. Therefore, this is the only proper normal nontrivial subgroup of $G$.

Edit. More precisely, $G_+$ is the maximal subgroup of $G$ whose every element acts on $T$ without inversions. This subgroup necessarily has index 2 in $G$. Equivalently, if we regard $T$ as a bipartite graph, then $G_+$ is the index 2 subgroup in $G$ preserving the parts of the vertex set.

$\endgroup$
6
  • 1
    $\begingroup$ One could also have a look at Serre's book "trees": springer.com/mathematics/algebra/book/978-3-540-44237-0 $\endgroup$
    – Ian Agol
    Feb 21, 2014 at 23:20
  • $\begingroup$ Thanks for the answer. But wouldn't it be easier to say that, given the unique 2-coloring of $T$, $G_+$ consists simply of the color-preserving automorphisms of $T$? $\endgroup$
    – Dominik
    Feb 24, 2014 at 20:58
  • $\begingroup$ @Dominik: Yes, it is equivalent, just the matter of taste. $\endgroup$
    – Misha
    Feb 24, 2014 at 22:07
  • $\begingroup$ The first given definition of $G_+$ is not correct. One correct definition of $G_+$ is that it is the subgroup of those $g$ such that for all $x$ we have $d(x,gx)\in 2\mathbf{Z}$. One other is that it is the set of $g$ preserving a 2-coloring (a map from vertices to $\{0,1\}$ such that adjacent vertices have different colors) - of course there are 2 2-colorings, not just 1, and this is why this subgroup has index 2. The coset $G-G^+$ contains loxodromic elements, which have no inversion. $\endgroup$
    – YCor
    Jan 16, 2018 at 1:23
  • 1
    $\begingroup$ To conclude, the famous notion of groups acting on a tree without inversion is not a natural notion. Essentially, it has only been coined so as to ensure that if there's a bounded orbit then there's a fixed vertex. The natural notion is the notion of bipartite-preserving action, i.e. an action through the group of index $\le 2$ of automorphisms preserving some bi-coloring. $\endgroup$
    – YCor
    Jan 20, 2018 at 3:08
1
$\begingroup$

Back in 1983, N. Gupta and S. Sidki (Math Z.) were constructing finitely generated infinite p-groups as groups of automorphisms of p-regular trees. There is a later treatment of their work by Gilbert Baumslag, but I don't have an immediate reference for his exposition. I think that Baumslag mostly looks at the special case where p=3. This work might be irrelevant to Dominik, but it might be of interest.

All trees are planar.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.