Let p ≥ 3 be a prime. A generalized multi-edge spinal group
$$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$
is a subgroup of the automorphism group of a regular
p-adic rooted tree
T that is generated by one rooted automorphism
a and
p families
$b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$
of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalizes the concepts of multi-edge spinal groups, including the widely studied GGS groups (named after Grigorchuk, Gupta and Sidki), and extended Gupta–Sidki groups that were introduced by Pervova [‘Profinite completions of some groups acting on trees,
J. Algebra 310 (2007), 858–879’]. Extending techniques that were developed in these more special cases, we prove: generalized multi-edge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore, we use tree enveloping algebras, which were introduced by Sidki [‘A primitive ring associated to a Burnside 3-group,
J. London Math. Soc. 55 (1997), 55–64’] and Bartholdi [‘Branch rings, thinned rings, tree enveloping rings,
Israel J. Math. 154 (2006), 93–139’], to show that certain generalized multi-edge spinal groups admit faithful infinite-dimensional irreducible representations over the prime field ℤ/
pℤ.