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My questions are:

Is there any group, which cannot be written as the quotient of a residually finite group by an amenable normal subgroup? Is it possible for large classes of groups?

and

Is there a group, such that every extension by an amenable group is split?

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1  
Could you give some motivation, and explain what (if anything) the two questions have to do with each other? –  HJRW Oct 4 '10 at 14:22
    
@Henry: None of these questions requires an additional motivation. @Andreas: I suspect that the free Burnside group of large enough exponent is an example. That would solve mathoverflow.net/questions/37344/… (see the discussion there). But that is a difficult question. –  Mark Sapir Oct 4 '10 at 16:08
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̯@Henry: If the second question has a positive answer, then this would (most likely) help to answer the first. My motivation is the study of sofic groups. In the sense of measured group theory, every sofic group is the quotient of a residually finite group by an amenable subgroup (but that takes some time to explain). I was wondering what kind of condition one gets in the more rigid setup of ordinary group theory. Burnside groups are not known to be sofic, so Mark's comment is very interesting. –  Andreas Thom Oct 4 '10 at 16:32
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I meant that no motivation is required because if the answer to the first question is "no", then it would be the first non-trivial property of all groups (which would disprove Gromov's thesis). And if the answer is "yes", it would provide a new way to construct groups because old ways don't seem to work. –  Mark Sapir Oct 4 '10 at 17:45
    
@Andreas: who proved the result of measured group theory that you mentioned, and where? –  Mark Sapir Oct 4 '10 at 17:55

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