Fix the wrong vertices number of Cayley graphs of free groups

[cxzhong]

fix #40549 Alternative #41173
Now, we only implement hash for finite groups and finite groups. They are well-defined behavior. for general infinite f.p. group, we can not decide word problem. I think setting they are unhashable is better

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

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[cxzhong]

@orlitzky Can you review this part for graph theory

[Copilot]

Pull request overview

This PR fixes incorrect vertex counts in Cayley graphs for finitely presented groups by addressing the word problem - the issue that equal group elements can have different word representations. The solution introduces a __hash__ method based on Tietze words and a specialized cayley_graph method that uses permutation group representations to correctly identify equal elements.

Key changes:

• Adds __hash__ method to FinitelyPresentedGroupElement based on Tietze representation (with documented limitations)
• Implements specialized cayley_graph method for FinitelyPresentedGroup that converts to permutation groups for proper element identification
• Adds cayley_graph methods to BraidGroup and ArtinGroup that delegate to the semigroup implementation

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 11 comments.

File	Description
src/sage/groups/finitely_presented.py	Adds __hash__ method to group elements and specialized cayley_graph method that uses permutation group conversion to correctly identify equal elements with different word representations
src/sage/groups/braid.py	Adds cayley_graph method that delegates to semigroup implementation, appropriate for infinite braid groups
src/sage/groups/artin.py	Adds cayley_graph method that delegates to semigroup implementation, appropriate for infinite Artin groups

Comments suppressed due to low confidence (3)

src/sage/groups/artin.py:689

• This method requires 1 positional argument, whereas overridden FinitelyPresentedGroup.as_permutation_group may be called with 2. This call correctly calls the base method, but does not match the signature of the overriding method.

    def as_permutation_group(self):

src/sage/groups/braid.py:2743

• This method requires 1 positional argument, whereas overridden FinitelyPresentedGroup.as_permutation_group may be called with 2. This call correctly calls the base method, but does not match the signature of the overriding method.

    def as_permutation_group(self):

src/sage/groups/finitely_presented.py:188

• This class implements hash, but does not implement eq.

class FinitelyPresentedGroupElement(FreeGroupElement):

---

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[orlitzky]

Whoever told you I know graph theory was lying :)

I've lost track of what was going on in #41173 but the discussion does not seem 100% dead yet.