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Linear and Muitiiinear Algebra. i979, Voi. 7, pp. 17719i
03081087:'79;0703417i SO4.5010
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
SL(2,R)generated by
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 ChangjenningsRee. 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.
