The problem of finding an upper bound for 
can be traced back to Schottky
[16].
Burnside
[5]
conjectured
;
this was disproved by Myrberg
[9],
[10].
Akaza
[1],
[2]
gave examples with
.
Sarnak
[15]
and Phillips
proved the existence of examples with 
.
Phillips and Sarnak
[13]
conjectured the existence of a universal upper bound 
,
and proved the analogous result in higher dimensions.
Brooks
[4],
[3]
proved the conjecture for the special class of groups
for which the disks
are a subset of the disks of the Apollonian packing.
(See Figure 2.)
Phillips, Sarnak, and Brooks have suggested that
the supremum of 
over all such ``Apollonian'' Schottky groups
should equal the supremum over all classical Schottky groups,
but this is not known.
Figure 2: The Apollonian packing.