Fundamental domains for congruence subgroups of SL2 in positive characteristic.
Abstract
Morgenstern ([M]) claimed to have constructed fundamental domains for
congruence subgroups of the lattice group Gamma=PGL_2(F_q[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 Gamma on the Bruhat-Tits
tree X of G=PGL_2(F_q((t^-1))). 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 SL_2(F_q[t]) and PGL_2(F_q[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.
Вход в систему
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