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4h
comment Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
Given that I found the idea of such a proof quite compelling, I figured others had the same though, and maybe someone had found one since the book was written.
4h
comment Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
@MartinBrandenburg I agree, the Andrews-Curtis conjecture, 3d Poincare conjecture, "is every hyperbolic group residually finite?", are only some of the research problems he has in the book. It has been 15 years or so since the book was published, so I am wondering is such a proof has been done, and I don't think at the time of the book being published there was such a proof. It is also really difficult to find papers on this, the only reason I found the ones sited above was because he sited them! (It is sometimes a little ambiguous as to "how solved" the research problems are)
7h
reviewed Looks OK Definite integral calculation
7h
reviewed Approve Constructing Simultaneous Equation for This Problem
8h
comment Open mathematical questions for which we really, really have no idea what the answer is
I have been under the impression that most think that there should be such a group, just that with current tech we don't have much of an idea to construct such a group (or actually carry out the ideas that could construct such a group). Are there group theorists that seriously believe that there is no such group, or some reason to believe there is none? Are their opinions the same for both bounded and unbounded exponent case?
14h
awarded  Nice Question
23h
revised Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
deleted 1 character in body
23h
comment Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
@LeeMosher I have added the ones I know of
23h
revised Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
Added sources for the results I am asking about.
1d
revised Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
Added some info, links, and a different question that could be answered, if the first does not have an answer at the moment.
1d
comment Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
I may be using the term "arithmetic dynamics" informally, as I don't reallly know much about the field.
1d
comment Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
@BrunoJoyal I get you point, so I changed it to say permutations. Although I do think part of the reason that it is interesting is that they are polynomials and so it feels like that there should be some nice action (possibly coming from some arithmetic dynamics perspective).
1d
revised Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
added 1 character in body
1d
asked Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
2d
answered Why is this group not free?
May
17
revised The ends of a group
Generally bad form for the full link to be posted like that, so improved formatting
May
17
comment The ends of a group
Did you read the previous couple sections on quasi-isometries? Consider the natural action of a finite index subgroup on a cayley graph of the full group...(apply some theorems from that section here)
May
14
reviewed No Action Needed Isomorphism of the Clifford bundle of a Riemannain manifold
May
14
reviewed No Action Needed Two stochastic variables
May
14
reviewed Close Linear transformation to evalute double integral