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) 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:

Does there exist an infinitely-ended finitely generated group whose space of ends is not Hölder-equivalent to the standard triadic Cantor set?

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).


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