(m,n)-graphs

How to build (m,n)-graphs

Let us consider the Baumslag-Solitar group BS(m,n) given by the presentation < t,b | tb^m = bⁿt >.

We describe below the local structure of the labeled oriented Bass-Serre graphs associated to right BS(m,n)-actions, as defined in [CGLS22, Sec. 3]. These are saturated (m,n)-graphs. Recall that they are obtained from Schreier graphs by

deleting the b-edges;
shrinking the vertices belonging to the same b-orbit to a single vertex labeled by the cardinality of the b-orbit;
shrinking the t-edges of the Schreier graph to a single edge when they start from points in the same bⁿ-orbit. The resulting positive edge is labeled by the cardinality of the bⁿ-orbit.

Note that we only described positive edges; by definition the negative edges are their opposites. The ingoing (resp. outgoing) degree of a vertex v is the number of positive edges ending at (resp. starting from) v.

If you take m = and n = , then we are working with BS(, ) and t sends b-orbits to b-orbits. The maximal ingoing degree is and the maximal outgoing degree is .

Consider a vertex v whose label is L(v) = . The phenotype of the label is = Ph_,().

The vertex v has = gcd(,) ingoing positive edges, with label , coming from vertices whose labels can be chosen (independently) in {}.

The vertex v has = gcd(,) outgoing positive edges, with label , going to vertices whose labels can be chosen (independently) in {}.

We mention separately the simpler case where the label is infinite: L(v) = ∞. Then the phenotype of the label is ∞ and all vertices and edges in the connected component of v have label ∞.

Every vertex in the connected component of v has positive ingoing edges and positive outgoing edges.

More generally, non-saturated (,)-graphs can be built by following the above constraints but allowing oneself to have less ingoing or outgoing positive edges than prescribed.

Finally, the case when m or n is negative is treated by noting that (m,n)-graphs are the same as (|m|,|n|)-graphs.

Reference

[CGLS22] A. Carderi, D. Gaboriau, F. Le Maître and Y. Stalder. On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype. preprint.