In [5, Lemma 5.1] we made the following claim. We claimed that if G is a locally compact group and is a compact open subgroup, then for all we have
As previously reported in , Yuhei Suzuki pointed out that this claim is not correct and provided the counterexample , where the action of flips the two copies of . The mistake in the proof of [5, Lemma 5.1] lies in the second equality of the first displayed equation, which exchanges a square of a sum with the sum of the squares.
It is a priori not clear whether the invalidity of [5, Lemma 5.1] affects any of the main results of . In this erratum, we show that the main example of a non-discrete -simple group obtained in Theorem G persists, while Theorem B is not correct in the stated generality.
In [5, Theorem G] we claimed that given the relative profinite completion of the Baumslag–Solitar group is -simple. We sketch an alternative proof for this statement, which does avoid the use of [5, Lemma 5.1].
Proof for Theorem G.
Since is a discrete Powers group if , it suffices to consider the case . Fix such m and n and write . Denote by the modular function and by the kernel of Δ. Denote by the image of the free letter. Then . If we show that is simple and has a unique tracial weight, then work of Archbold and Spielberg [1, Corollary, p. 122] shows that also is simple, since the action on scales the Plancherel trace by .
Denote by the elliptic subgroup of G, and recall that G is an HNN-extension of K by . Denote by the associated t-exponent sum, as used for example in . Denote by
the t-height of . It is the absolute value of the maximal t-exponent sum of a terminal piece of w. We put
Then is a an open subgroup of and is a normal compact open subgroup of . Since , it suffices to show that is -simple by Suzuki’s result .
Denote by the Bass–Serre tree of G, and write ρ for its root. Write and observe that is a subtree. Note that the action of on is trivial, so that we obtain a well-defined action of on . We observe that for every non-empty open subset there are infinitely many hyperbolic elements in with pairwise different endpoints, which all lie in O. So by the arguments of , the group is a Powers group and thus -simple, if we can exhibit for every finite subset a non-empty open subset such that . To this end, it suffices to show that the set of fix points of every non-trivial element of is meager in . Hyperbolic elements have at most two fixed points, while the action of any non-trivial elliptic element in is conjugate to one of the elements , whose fixed points lie in . Since the latter subset is meager, we can conclude that G is -simple. ∎
As explained above, [5, Theorem B] is not correct as stated. A counterexample is provided by the Burger–Mozes group associated with the permutation group with its imprimitive action on a set with six elements. This counterexample as well as the following criterion for non- -simplicity were suggested to us by Pierre-Emmanuel Caprace. We thank him for allowing us to include them into this piece. An action of a group G on a space X is called micro-supported if for all non-empty open subsets there is some acting non-trivially on X and fixing pointwise. Note that the action of any Burger–Mozes group on the boundary of its tree is micro-supported, thanks to Tits’ independence property.
Let G be a locally compact group acting on a compact space X. Assume that the action of G has some open amenable point stabiliser and is micro-supported. Then G is not -simple.
We may assume that X is non-trivial, since amenable groups are not -simple. Let have an open and amenable point stabiliser. Then is unitarily equivalent with the quasi-regular representation on and thus weakly contained in the left-regular representation. Denote by the associated -homomorphism. Since the inclusion is essential, it suffices to show that the extension of π to the multiplier algebra is not injective. Let be two disjoint open subsets. Let be elements acting non-trivially on X such that g fixes pointwise and h fixes pointwise. Then . Indeed, calculating with the orthogonal basis of , we find if and if . In the latter case, we have . So that follows for all . ∎
 R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete -dynamical systems, Proc. Edinb. Math. Soc. (2) 37 (1994), no. 1, 119–124. 10.1017/S0013091500018733Search in Google Scholar
 P. de la Harpe and J.-P. Préaux, -simple groups: Amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds, J. Topol. Anal. 3 (2011), no. 4, 451–489. 10.1142/S1793525311000659Search in Google Scholar
 M. Elder and G. Willis, Totally disconnected groups from Baumslag–Solitar groups, Infinite group theory. From the past to the future, World Scientific, Singapur (2018), 51–79. 10.1142/9789813204058_0004Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.