Linear and Muitiiinear Algebra. i979, Voi. 7, pp. 177-19i
C Gordon z d Breach Science Publishzrs; Iiic.; 1979
Printed in Great %riain
In 1947 Sanov [S] proved that G(2) is free. Some years later, Brenner [Br]
showed that G(m)is free for all Iml g 2 and Chang, Jennings and Ree [CJR]
showed that values of m for which G(n?)is not even torsion free are dense io
rl. ' * - -LLICiiitC?~v'di -L, 21.
Xccentiy, Bachrnuth and Xvchizuki jBMj defined subgroups Gja, P, of
sho:wd that f aa, ,R, y 1 4.45 G(c,k, y) is free, and asked whether G(E,3, y)
can be contained in any free subgroup of rank 2.
In Section 1 we present preliminary notions and quick geometric proofs of
these theorems of Sanov, Brenner and Chang-jennings-Ree. Much more is
known about these groups G(mj and other subgroups of Si(2,R) of rank
two than -theseresults (see for example [A'],[LZi,],[LU,]).Our intention here
is to recapitulate only that part which is geometrically obvious and which
motivates our iln.&sir of the -E ~ C ? ! I ~ C- G(,y;B; 7 ) .
Supported in part by an NSF Grant.