[#P2908] A free group generated by exponential and squaring maps

Problem. On \((0,\infty)\), let \(f(x)=e^x-1\) and \(g(x)=x^2\). Do the homeomorphisms \(f\) and \(g\) generate a free group of rank two under composition?

An illustration accompanying the statement of A free group generated by exponential and squaring maps. — An illustration accompanying the statement. The statement alone defines the problem.

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Definitions and notation

1Context

Normal forms, asymptotic expansions near zero or infinity, and certified separating points for word families are reusable. A proof must control arbitrary word length, where cancellations between logarithms, exponentials, squares, and roots become difficult.

2Problem setup

Remark 1. The generated group includes f^{-1}(x)=log(1+x) and g^{-1}(x)=sqrt(x), with the positive square root.

Definition 1. The group is free of rank two if every nonempty freely reduced word in f, f^{-1}, g, and g^{-1} acts as a nonidentity homeomorphism of (0,infinity).

3What counts as a solution

Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0.
A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The sole answer proposes a consequence of Schanuel's conjecture. Comments identify gaps in how reduced words enter the argument and ask whether Ax-Schanuel yields an unconditional proof; the thread contains no settled unconditional resolution. Exact unresolved remainder: Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.^[3][2][1]

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The sole answer proposes a consequence of Schanuel's conjecture. Comments identify gaps in how reduced words enter the argument and ask whether Ax-Schanuel yields an unconditional proof; the thread contains no settled unconditional resolution. Exact unresolved remainder: Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The sole answer proposes a consequence of Schanuel's conjecture. Comments identify gaps in how reduced words enter the argument and ask whether Ax-Schanuel yields an unconditional proof; the thread contains no settled unconditional resolution. Exact unresolved remainder: Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.^[3][2][1] Relevance to this problem For A free group generated by exponential and squaring maps, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 3, Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The sole answer proposes a consequence of Schanuel's conjecture. Comments identify gaps in how reduced words enter the argument and ask whether Ax-Schanuel yields an unconditional proof; the thread contains no settled unconditional resolution. Exact unresolved remainder: Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target. Open full record →
Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.^[1] Relevance to this problem For exponential square free group, sets the next completion test: Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-27. The sole answer proposes a consequence of Schanuel's conjecture. Comments identify gaps in how reduced words enter the argument and ask whether Ax-Schanuel yields an unconditional proof; the thread contains no settled unconditional resolution.

Ax's 1971 theorem on Schanuel-type differential results was checked because the thread asks whether it applies. The audit did not find a published specialization proving this exact homeomorphism-group statement.
Searches for the two generators, their inverse maps, and free subgroups of Homeo_+(R) found general constructions but no direct unconditional treatment of this pair.
A local TheoremDB search for exponential-square generators and the exact functional words found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. The commutator f composed with g composed with f^{-1} composed with g^{-1} is visibly nonidentity at many numerical test points, but checking finitely many words cannot prove freeness.

2See also

The sharp Dirichlet Poincare constant with two moment constraintsreal analysis
Optimal symmetric three-point quadrature for a bounded second derivativereal analysis
Decidability of the real exponential fieldreal analysis

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Plain text

“A free group generated by exponential and squaring maps.” TheoremDB. P2908. Problem statement; statement text SHA-256 7a7a733da86c497799f219382b4b568ad3e4d567bb305b23a96eaa72ef890663. https://theoremdb.org/statement/?ref=P2908

BibTeX

@misc{theoremdb-problem-7a7a733da86c497799f219382b4b568ad3e4d567bb305b23a96eaa72ef890663,
  title = {{A free group generated by exponential and squaring maps}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 7a7a733da86c497799f219382b4b568ad3e4d567bb305b23a96eaa72ef890663},
  url = {https://theoremdb.org/statement/?ref=P2908}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. MathOverflow question 499819, “A free group generated by exponential and squaring maps,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 499819, its answer, and every visible comment were checked on 2026-07-27.Also cited at Full question, answers, and visible comments concerning A free group generated by exponential and squaring maps; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For A free group generated by exponential and squaring maps, the reviewed source scope is Full question, answers, and visible comments concerning A free group generated by exponential and squaring maps; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
James Ax, “On Schanuel's Conjectures,” Annals of Mathematics 93(2) (1971), 252. DOI 10.2307/1970774. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at main differential-field Schanuel theorem.Source used to assess the problem's recorded status.For A free group generated by exponential and squaring maps, this source supplies the Ax-Schanuel-type tool mentioned by the source discussion; it does not resolve the concrete generated group.
Published article identified by DOI 10.1080/00927872.2010.489918, linked primary source for “A free group generated by exponential and squaring maps.”. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and results on algebraic roots of exponential polynomials under Schanuel's conjecture.Source used to assess the problem's recorded status.For A free group generated by exponential and squaring maps, this source supplies conditional exponential-algebra background for the proposed route, without proving freeness of the two self-maps.

An original CC0 reformulation motivated by the cited MathOverflow discussion; no wording was copied from the page.

[#P2908] A free group generated by exponential and squaring maps

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