Abstract. There are 432 strongly squarefree symmetric bilinear forms of signature (2, 1) defined over Z[ √ 2] whose integral isometry groups are generated up to finite index by finitely many reflections. We adapted Allcock's method (based on Nikulin's) of analysis for the 2-dimensional Weyl chamber to the real quadratic setting, and used it to produce a finite list of quadratic forms which contains all of the ones of interest to us as a sub-list. The standard method for determining whether a hyperbolic reflection group is generated up to finite index by reflections is an algorithm of Vinberg. However, for a large number of our quadratic forms the computation time required by Vinberg's algorithm was too long. We invented some alternatives, which we present here.
In this short note we use the presentations found by the various authors to show that the Picard modular groups
PU
(
2
,
1
,
O
d
)
\operatorname {PU}(2,1,\mathcal {O}_d)
with
d
=
1
,
3
,
7
d=1,3,7
(respectively the quaternion hyperbolic lattice
PSp
(
2
,
1
,
H
)
\operatorname {PSp}(2,1,\mathcal {H})
with entries in the Hurwitz integer ring
H
\mathcal {H}
) are generated by complex (resp. quaternionic) reflections, and that the Picard modular groups
PU
(
2
,
1
,
O
d
)
\operatorname {PU}(2,1,\mathcal {O}_d)
with
d
=
2
,
11
d=2,11
have an index 4 subgroup generated by complex reflections.
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