Quotient graphs for certain arithmetic subgroups of PGL_3 over a rational function field

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ChristBastian_2018_07_10.pdf (1.06 MB)

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2018

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http://dx.doi.org/10.22029/jlupub-9781

Abstract

In the present thesis we consider the action of PGL_3(F_q(t)), where F_q(t) is the rational function field over the finite field with q elements, on the associated Bruhat-Tits building. For a place p of F_q(t) there is the subring O_{p} of the rational function field consisting of those rational functions having possible poles only at p. We compute the quotient graph for the action of the arithmetic subgroup PGL_3(O_{p}) on the underlying graph (i.e. the 1-skeleton) of the Bruhat-Tits building corresponding to the place p.Therefore we first compute the fundamental domain for the action of PGL_3(F_q[t]) on the underlying graph of the Bruhat-Tits building associated to the place infinity. Then we consider the action of PGL_3(O_{p,infinity}), where O_{p,infinity} are those rational functions with possible poles only at p and at infinity, on the product of the Bruhat-Tits building associated to p and the Bruhat-Tits building associated to infinity. The PGL_3(O_{p})-orbits on the vertices of the Bruhat-Tits building corresponding to p can be identified with certain PGL_3(F_q[t])-orbits on the vertices of the Bruhat-Tits building associated to infinity. So we can describe the vertices of the quotient graph. For the number of edges between two given vertices in the quotient we can compute the number of corresponding double cosets. To actually compute the number of double cosets we have to distinguish a lot of cases, so that we can calculate the needed cardinalities of the involved sets in order to find the number of double cosets in each of these cases. In the Main Theorem we describe the quotient graphs.

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