Fundamental Domains for Congruence Subgroups of Sl2 in Positive Characteristic

Morgenstern ([Mor95]) claimed to have constructed fundamental domains for congruence subgroups of the lattice group Γ = PGL2(Fq[t]), and subgraphs providing the first known examples of linear families of bounded concentrators. His method was to construct the fundamental domain for a congruence subgroup as a ‘ramified covering’ of the fundamental domain for Γ on the Bruhat-Tits tree X = Xq+1 of G = PGL2(Fq((t ))). We prove that Morgenstern’s constructions do not yield the desired ramified coverings, and in particular yield graphs that are not connected in characteristic 2. It follows that Morgenstern’s graphs cannot be quotient graphs by the action of congruence subgroups on the Bruhat-Tits tree. Moreover, subgraphs of Morgenstern’s graphs which he claims to be expanders are also not connected. We clarify the construction of Morgenstern and we prove that his full graphs are connected only in odd characteristic. We also repair his constructions of ramified coverings. We construct fundamental domains for congruence subgroups of SL2(Fq[t]) and PGL2(Fq[t]) as ramified coverings, and we give explicit graphs of groups for a number of congruence subgroups. We thus provide new families of graphs whose level 0 − 1 subgraphs potentially have the expansion properties claimed by Morgenstern.