Virtual isomorphism types of the embeddings φ_{1,m}

Determine the virtual isomorphism types, in the sense of Bader–Furman–Sauer, of the lattice embeddings φ_{1,m}: BS(1,n^l) → G_n constructed in the paper, where φ_{1,m} maps the generator a of BS(1,n^l) to (a_1)^m and b^l to b^l under the standard action on X_n. Provide a classification of these embeddings up to virtual isomorphism (i.e., up to passing to finite index subgroups and compact kernels/quotients).

Background

In the context of classifying lattice envelopes, Bader–Furman–Sauer show that, for broad classes of lattices, embeddings into locally compact groups are classified up to virtual isomorphism. The group G_n studied in this paper can be realized as a tree extension, placing it close to that framework.

The paper classifies lattice embeddings BS(1,n^l) → G_n up to conjugacy and describes Aut(G_n). However, the authors note that they lack a description of the virtual isomorphism types of the specific embeddings φ_{1,m}, leaving open how these embeddings relate under the weaker equivalence of virtual isomorphism.