The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization ([wikipedia link](https://en.wikipedia.org/wiki/Stallings_theorem_about_ends_of_groups)) in terms of amalgam/HNN splittings over finite subgroups. Up to homeomorphism, this provides a complete picture. However, the space carries a canonical Hölder structure, described below. My question is:



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> Does there exist an infinitely-ended finitely generated group whose space of ends is not Hölder-equivalent to the standard triadic Cantor set?
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Given a connected locally finite graph and ends $\eta,\omega$, a base-point $x$ and $B\_x(n)$, define $v\_x(\eta,\omega)$ as the largest $n$ such that there a connected component of $X\smallsetminus B\_x(n)$ accumulating on both $\eta$ and $\omega$ (this equals $\infty$ iff $\eta=\omega$). This yields an ultrametric $d\_x=2^{-v\_x}$ on the space of ends. (If you're familiar to the Gromov boundary of Gromov-hyperbolic spaces, you may somewhat recognize this as an easier analogue.) The bilipschitz type of the boundary does not depend on $x$; more precisely the identity map $(X,d\_x)\to (X,d\_y)$ is bilipschitz for all $x,y\in X$. Furthermore, it is easy to see that quasi-isometries between such graphs induce Hölder homeomorphisms between space of ends. In particular, for a finitely generated group, this Hölder structure on the space of ends is well-defined. 


For the more obvious examples I can imagine (say, virtually free groups), this yields the standard Hölder structure on the Cantor space. But I do not know in general, even assuming that the group is accessible.


For graphs whose space of ends is homeomorphic to a Cantor space, it is an instructive exercise to produce nonstandard Hölder structures on the space of ends (take a triadic regular tree and assign a huge length to some edges).



Finitely generated groups with Hölder-exotic space of ends?
The question is solved positively in the paper *The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups* by Matthieu Dussaule, Wenyuan Yang, which appeared on arXiv on October of 2020 ([link](https://arxiv.org/abs/2010.07671)). It even says that this is almost always the case. This means that the space of ends, just endowed with its metric structure "up to Hölder equivalence", is an interesting quasi-isometry invariant of infinitely-ended groups.


Namely Corollary 1.9 therein says



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> Let $\Gamma$ be an infinitely-ended finitely generated group. Suppose that $\Gamma$ is accessible [e.g., $\Gamma$ is finitely presented, or virtually torsion-free]. Then the end boundary of $\Gamma$ is bi-Hölder equivalent to the
> standard Cantor ternary set if and only if $\Gamma$ is virtually free.
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For instance, the end boundary of the free product $\mathbf{Z}^2\ast\mathbf{Z}$ is not Hölder-equivalent to the
standard Cantor.


