7
$\begingroup$

I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.

I am reading this thesis.

Corollary 4.1.15. on page 63 says that if the number of ends (see Definition 4.1.6. of the above thesis) of Freudenthal space (i.e., a $\sigma$-compact, locally compact, connected, and locally connected space) is infinite, then its space of ends is homeomorphic to the Cantor set.

As far as know, the space of ends of a non-compact surface can be any closed subset of the Cantor set; see Theorem 2 of this. For example, the space of ends of $\Bbb R^2\setminus \Bbb N$ is homeomorphic to the space $\{1/n\}\cup\{0\}$, which is an infinite set but certainly not homeomorphic to the Cantor set.

So, my question is the following:

Are these two notions of the space of ends (one coming from Freudenthal compactification and another, as mentioned in Ian Richard's paper) different?

Improve this question
$\endgroup$
1
  • 7
    $\begingroup$ No, there's only one notion of ends for such spaces, and the Corollary 4.1.15 of this thesis is plainly false. The second sentence in the proof sounds suspicious (apply it to an isolated end in any counterexample to see that the proof can't work?). $\endgroup$
    – YCor
    Dec 5, 2022 at 21:30

1 Answer 1

Reset to default
2
$\begingroup$

Of course, yourself and YCor already answered this in the comments, but since I see you tagged geometric group theory, maybe you will be interested in this explicit answer about bi-Holder homeomorphisms : The end boundary of an accessible infinitely-ended group is bi-Hölder equivalent to the standard Cantor ternary set if and only if it is virtually free. This is Corollary 1.9 of this paper : https://arxiv.org/abs/2010.07671. The involved distance is the visual distance with respect to a base-point.

Improve this answer
$\endgroup$
3
  • 2
    $\begingroup$ This has little to do with this question here. But this is precisely the answer to this other question (of mine, 4 years ago) and I already posted as a cw answer a link to your paper — which I noticed more than 2 years after it was posted on arXiv. $\endgroup$
    – YCor
    Dec 5, 2022 at 23:32
  • 1
    $\begingroup$ @YCor You're right, but I don't get why the OP tagged geometric group theory, if they only look for the homeomorphism type. If this really has nothing to do with the OP intentions in this question, I'll delete my answer. $\endgroup$
    – M. Dus
    Dec 6, 2022 at 0:20
  • 1
    $\begingroup$ Sure, but should this answer could be posted to any question about ends of groups? There are many such questions. $\endgroup$
    – YCor
    Dec 6, 2022 at 8:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.