# P2908: A free group generated by exponential and squaring maps

- ID: `P2908`
- Reference: `exponential-square-free-group`
- Page: https://theoremdb.org/statements/P2908
- Record maturity: Reviewed problem with recorded work

## 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?

### Problem setup

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

### What 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.

## Status

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](#reference-3) [2](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- 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.
- The MathOverflow question, answer, and all comments were checked on 2026-07-27. The visible argument is conditional on Schanuel's conjecture, and commenters dispute whether its algebraic-independence step handles every reduced word.
- 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.

- Recorded example: 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `exponential-square-free-group`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 499819, “A free group generated by exponential and squaring maps,” checked 2026-08-01. Question 499819, its answer, and every visible comment were checked on 2026-07-27. https://mathoverflow.net/questions/499819/do-x-mapsto-ex-1-and-x-mapsto-x2-generate-a-free-group-on-mathbbr-to
   - 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.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>James Ax, “On Schanuel's Conjectures,” Annals of Mathematics 93(2) (1971), 252. DOI 10.2307/1970774. main differential-field Schanuel theorem https://doi.org/10.2307/1970774
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Ahuva C. Shkop, “Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials,” Communications in Algebra 39(10) (2011), 3813-3823. DOI 10.1080/00927872.2010.489918. abstract and results on algebraic roots of exponential polynomials under Schanuel's conjecture https://doi.org/10.1080/00927872.2010.489918
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.