$C^*$-irreducibility of commensurated subgroups

Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the inclusion $\Lambda\leq \Gamma$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that $\rm{PSL}(n,\mathbb{Z})\leq\rm{PGL}(n,\mathbb{Q})$ is $C^*$-irreducible for any $n\in \mathbb{N}$, and that the inclusion of a $C^*$-simple group into its abstract commensurator is $C^*$-irreducible. The main ingredient that we use is the fact that the action of a commensurated subgroup $\Lambda\leq\Gamma$ on its Furstenberg boundary $\partial_F\Lambda$ can be extended in a unique way to an action of $\Gamma$ on $\partial_F\Lambda$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.


Introduction
A group Γ is said to be C * -simple if its reduced C * -algebra C * r (Γ) is simple. After the breakthrough characterizations of C * -simplicity in [KK17] and [BKKO17], several directions of research applying the new methods in different settings arose.
One of the recent interesting directions is investigating when inclusions of groups Λ ≤ Γ are C * -irreducible, in the sense that every intermediate C * -algebra B in C * r (Λ) ⊂ B ⊂ C * r (Γ) is simple. In [Rør21], Rørdam started a systematic study of this property and provided a dynamical criterion for an inclusion of groups to be C * -irreducible. Together with results in [Amr21], [Urs22] and [BO23], this has provided a complete characterization of C * -irreducibility of an inclusion in the case that Λ is a normal subgroup of Γ.
Recall that a subgroup Λ of a group Γ is said to be commensurated if, for any g ∈ Γ, Λ ∩ gΛg −1 has finite index in Λ. This is a much more flexible generalization of normal subgroups and finite-index subgroups. For example, for every n ≥ 2, PSL(n, Z) is an infinite-index commensurated subgroup of the simple group PSL(n, Q).
In this work, we generalize the above characterization of C * -irreducibility to commensurated subgroups (see Theorem 3.5). The main ingredient in our proof is the fact that the action of Λ on its Furstenberg boundary ∂ F Λ can be uniquely extended to an action of Γ on ∂ F Λ if Λ is a commensurated subgroup in Γ (see Theorem 3.1).
As one of the applications, we show that, if Γ is a C * -simple group, then the inclusion of Γ in its abstract commensurator Comm(Γ) is C * -irreducible (see Corollary 3.14). To our best knowledge, this is also the first observation of the fact that, if Γ is a C * -simple group, then Comm(Γ) is C * -simple as well.
Given a subgroup Λ of a group Γ, Ursu introduced in [Urs22] a universal Λstrongly proximal Γ-boundary B(Γ, Λ) and showed that, if Λ Γ, then B(Γ, Λ) = This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 817597). ∂ F Λ. In Section 4, we generalize this fact to commensurated subgroups and also observe that, in general, B(Γ, Λ) is not extremally disconnected.
Finally, we also show that, given a commensurated subgroup Λ of a group Γ, the action of Λ on its universal minimal proximal space ∂ p Λ can also be extended in a unique way to an action of Γ on ∂ p Λ (see Theorem 5.1), and use this fact for concluding that, for a certain locally finite commensurated subgroup G of Thompson's group V , the resulting action of V on ∂ p G is free (see Example 5.4).

Preliminaries
Given a compact Hausdorff space X, we denote by Prob(X) the space of regular probability measures on X. An action of a group Γ on X by homeomorphisms is said to be minimal if X does not contain any non-trivial closed invariant subset, and to be topologically free if, for any g ∈ Γ \ {e}, the set {x ∈ X : gx = x} has empty interior (if Γ is countable, then Γ X is topologically free if and only if the set of points in X which are not fixed by any non-trivial element of Γ is dense in X). The action is said to be proximal if, given x, y ∈ X, there is a net (g i ) ⊂ Γ such that the nets (g i x) and (g i y) converge and lim g i x = lim g i y. We say that the action is strongly proximal if the induced action Γ Prob(X) is proximal. The action is called a boundary action (or X is a Γ-boundary) if it is both minimal and strongly proximal. We denote by ∂ F Γ the Furstenberg boundary of Γ, i.e., the universal Γ-boundary (see [Gla76, Section III.1]). The group Γ is C * -simple if and only if Γ ∂ F Γ is free ([BKKO17, Theorem 3.1]).
Given Γ-boundaries X and Y , if there exists ϕ : X → Y a homeomorphism which is Γ-equivariant (Γ-isomorphism), then it follows from [Gla76, Lemma II.4.1] that ϕ is the unique Γ-isomorphism between X and Y .
Given a group isomorphism ψ : Γ 1 → Γ 2 , by universality there is a unique homeomorphismψ : Given a group Γ, let Sub(Γ) be the space of subgroups of Γ endowed with the pointwise convergence topology and with the Γ-action given by conjugation. Given a subgroup Λ ≤ Γ, a Λ-uniformly recurrent subgroup (URS) is a non-empty closed Λ-invariant minimal set U ⊂ Sub(Γ). Moreover, we say that U is amenable if one (equivalently all) of its elements is amenable. By [Ken20, Theorem 4.1], a group Γ is C * -simple if and only if it does not admit any non-trivial amenable Γ-uniformly recurrent subgroup.
An inclusion of groups Λ ≤ Γ is said to be C * -irreducible if every intermediate C * -algebra of C * r (Λ) ⊂ C * r (Γ) is simple. Given Λ ≤ Γ and g ∈ Γ, let g Λ := {hgh −1 : h ∈ Λ}. We say that Γ is icc relatively to Λ if, for any g ∈ Γ \ {e}, |g Λ | < ∞. The group Γ is said to be icc if it is icc relatively to itself.
A subgroup Λ ≤ Γ is said to be commensurated if, for any g ∈ Γ, Λ is commensurable with gΛg −1 . Equivalently, for any g ∈ Γ, [Λ : Λ ∩ gΛg −1 ] < ∞. In this case, we write Λ ≤ c Γ. In the literature, this notion is also referred to by saying that Λ is an almost normal subgroup of Γ or that (Γ, Λ) is a Hecke pair. The Finally, given g ∈ Λ, we have that Remark 3.2. The existence part of Theorem 3.1 was shown by Dai and Glasner in [DG19, Theorem 6.1] using a different method and assuming that Γ is countable.
Given a subset S of a group Γ, let C Γ (S) be the centralizer of S in Γ. In the next result, we follow the argument of [BKKO17, Lemma 5.3].
The proof of the following result is an adaptation of the argument in [Ken20, Remark 4.2] and its hypothesis is the same as in [Rør21, Theorem 5.3.(ii)].
Proposition 3.4. Let Λ ≤ Γ. Suppose that there exists a Γ-boundary X such that, for any µ ∈ Prob(X), there exists a net (g i ) ⊂ Λ such that g i µ converges to δ x , for some x ∈ X, on which Γ acts freely. Then Γ does not admit any non-trivial amenable Λ-URS.
Proof. Suppose U is a non-trivial amenable Λ-URS, and take K ∈ U. Since K is amenable, there exists µ ∈ Prob(X) fixed by K. Let (g i ) ⊂ Λ be a net such that g i µ → δ x , for some x ∈ X, on which Γ acts freely. By taking a subnet, we may assume that Theorem 3.5. Let Λ ≤ c Γ. The following conditions are equivalent: ( (1) =⇒ (2) follows from [Rør21, Remark 3.8 and Proposition 5.1].
(5) =⇒ (2). If Λ is not C * -simple, then it contains a non-trivial amenable Λuniformly recurrent subgroup. If Γ is not icc relatively to Λ, there exists s ∈ Γ \ {e} such that s Λ is finite. Hence the Λ-orbit of s is a finite non-trivial amenable Λ-uniformly recurrent subgroup.
Remark 3.6. In [Rør21, Theorem 5.3], Rørdam showed that an inclusion Λ ≤ Γ satisfying the hypothesis of Proposition 3.4 is C * -irreducible, and asked whether the converse holds. We do not know whether the converse of Proposition 3.4 holds and whether the absence of non-trivial amenable Λ-URS of Γ is equivalent to Λ ≤ Γ being C * -irreducible in general. Let (e ij ) 1≤i,j≤n ∈ M n (Z) be the matrix units and fix [a] ∈ PGL(n, Q) \ {[Id]}. By taking conjugates of [a] by elements of the form [Id+ m·e ij ] ∈ PSL(n, Z), m ∈ Z, 1 ≤ i = j ≤ n, it is easy to see that [a] PSL(n,Z) is infinite, so that PGL(n, Q) is icc relatively to PSL(n, Z).
The conclusion then follows from Theorem 3.5.
Remark 3.8. Let us sketch a different proof of Corollary 3.7 which gives the stronger statement that PSL(n, Z) ≤ PGL(n, R) is C * -irreducible, where PGL(n, R) is seen as a discrete group. Clearly, it suffices to show that, for any countable group Γ such that PSL(n, Z) ≤ Γ ≤ PGL(n, R), the inclusion PSL(n, Z) ≤ Γ is C * -irreducible. By the argument in [Bry17, Example 3.4.3], the action of PGL(n, R) on the projective space P n−1 (R) is topologically free. Since PSL(n, Z) P n−1 (R) is a boundary action, the result follows from [Rør21, Theorem 5.3].
Example 3.10. The inclusion given by the Sanov subgroup F 2 ≤ PSL(2, Z) is finite-index, hence it is C * -irreducible by Corollary 3.9.
Free groups. Fix m, n ∈ N such that 2 ≤ m < n and consider the free groups F m = a 1 , . . . , a m ≤ a 1 , . . . , a n = F n . In [Rør21, Example 5.4], Rørdam observed that F m ≤ F n is C * -irreducible. Notice that F m is far from being commensurated in F n . In fact, given g ∈ F n \ F m , we have that F m ∩ gF m g −1 = {e} (i.e., F m is malnormal in F n ). In particular, this example is not covered by Theorems 3.1 and 3.5. Nonetheless, there does exist an extension to F n of the action F m ∂ F F m , but it is far from being unique, since the generators a m+1 , . . . , a n can be mapped into any homeomorphisms on ∂ F F m .
Furthermore, we claim that F m ≤ F n satisfies condition (5) in Theorem 3.5. We will prove this by using Proposition 3.4. Let be the Gromov boundary of F n , and consider the action of F n on ∂F n by left multiplication. Fix µ ∈ Prob(∂F n ) and we will show that there is w ∈ ∂F n on which F n acts freely and such that δ w ∈ F m µ. Let z + := (a 1 ) i∈N ∈ ∂F n and z − := (a −1 1 ) i∈N ∈ ∂F n . Notice that, for all y ∈ ∂F n \ {z − }, we have that, as k → +∞, a k 1 y → z + . Furthermore, a 1 fixes z − . It follows from the dominated convergence theorem that as k → +∞. In particular, ν := µ({z − })δ z− + (1 − µ({z − })δ z+ ∈ F n µ. Let w := a 1 a 1 2 a 1 a 2 2 a 1 a 3 2 . . . a 1 a l 2 a 1 a l+1 2 · · · ∈ ∂F n . Since w is not eventually periodic, we have that F n acts freely on w. Given k ∈ N, let g k := w 1 . . . w k a 2 ∈ F m . We have that g k z ± = w 1 . . . w k a 2 z ± → w, as k → +∞. Therefore, δ w ∈ F m ν ⊂ F m µ, thus showing the claim.
Abstract commensurator. Let Γ be a group and Ω be the set of isomorphisms between finite-index subgroups of Γ. Given α, β ∈ Ω, we say that α ∼ β if there exists a finite-index subgroup H ≤ dom(α) ∩ dom(β) such that α| H = β| H . Recall that the abstract commensurator of Γ, denoted by Comm(Γ), is the group whose underlying set is Ω/∼, with product given by composition (defined up to finite-index subgroup).
Let Λ be a commensurated subgroup of Γ. Given g ∈ Γ, let and j Γ Λ : Γ → Comm(Λ) be the homomorphism given by j Γ Λ (g) := [β g ]. In order to ease the notation, we will sometimes denote j Γ Λ simply by j, and it will always be clear from the context what are the involved groups. Let us now collect a few elementary facts about j.
Lemma 3.11. Let Γ be a group.
Lemma 3.13. If Γ is an icc group, then Comm(Γ) is icc relatively to Γ.
Proof. Recall that any C * -simple group is icc (this follows, e.g., from Theorem 3.5). The result is then a consequence of Theorem 3.5 and Lemma 3.13.
Remark 3.16. Let F n be a non-abelian free group of finite rank. Then Corollary 3.14 implies that Comm(F n ) is C * -simple. In particular, it does not admit any non-trivial amenable normal subgroup. It is an open problem whether Comm(F n ) is a simple group ([CM18, Problem 7.2]).
Consider Γ := PSL(2, Z) and the boundary action Γ R ∪ {∞}. The stabilizer Γ ∞ of ∞ is isomorphic to Z and consists of the translations g n (x) := x + n, n ∈ Z, x ∈ R.
Remark 4.2. Let Γ be a group. One of the key properties in the applications of ∂ F Γ to C * -simplicity of Γ is the fact that C(∂ F Γ) is injective, shown in [KK17, Theorem 3.12]. Proposition 4.1 implies that C(B(Γ, Λ)) is not injective, in general. We believe that this is an evidence that B(Γ, Λ) is not likely to play the same role of the Furstenberg boundary in C * -algebraic applications.
Our next aim is to show that, given Λ ≤ c Γ, it holds that B(Γ, Λ) = ∂ F Λ. We start with a result which we believe has its own interest.
Theorem 4.3. Let Λ ≤ c Γ and Γ X a minimal action on a compact space such that Λ X is proximal. Then Λ X is minimal as well.
Proof. Let M ⊂ X be a closed non-empty Λ-invariant set. For any g ∈ Γ, we have that gM is gΛg −1 -invariant.
The following is an immediate consequence of the previous theorem: Corollary 4.4. Let Λ ≤ c Γ. If X is a Γ-boundary which is also Λ-strongly proximal, then X is a Λ-boundary.
By arguing as in [Urs22,Corollary 4.3], we conclude the following: One can easily check that the statements of Theorem 3.1 and Lemma 3.3 hold with ∂ p Λ instead of ∂ F Λ, with the exact same proofs (in particular, [BKKO17, Lemma 5.1], which is needed in the proof of Lemma 3.3, uses only proximality). Thus, we obtain:
(4) =⇒ (1). Suppose that there is g ∈ Γ \ {e} such that |g Λ | < ∞. Then H := {h ∈ Λ : gh = hg} is a finite-index subgroup of Λ, hence H ∂ p Λ is also minimal and proximal. Since the homeomorphism on ∂ p Λ given by g is Hequivariant, we conclude that g acts trivially on ∂ p Λ. Let us now apply Theorem 5.2 to a certain locally finite commensurated subgroup of Thompson's group V .
Example 5.4. Let X := {0, 1} and, given n ≥ 0, let X n be the set of words in X of length n. Given w ∈ X n , let C(w) := {(s n ) ∈ X N : s [1,n] = w}. Recall that Thompson's group V is the group of homeomorphisms on X N consisting of elements g for which there exist two partitions {C(w 1 ), . . . C(w m )} and {C(z 1 ), . . . , C(z m )} of {0, 1} N such that g(w i s) = z i s for every 1 ≤ i ≤ m and s ∈ X N .
Let us define inductively groups G n acting by permutations on X n . Let G 1 := Z 2 acting non-trivially on X and, for n ∈ N, where the action of G n+1 on X n+1 is defined as follows: given v ∈ X n , x ∈ X, σ ∈ G n and f ∈ X n Z 2 , (f, σ)(vx) := σ(v)f σ(v) (x).
Let G := lim n∈N G n . Then G acts faithfully on X N and, as observed in [LB17, Proposition 7.11], G ≤ c V .
We claim that V is icc relatively with G. Given u ∈ X n , let the rigid stabilizer of u, denoted by rist G (u), be the subgroup of G consisting of the elements which, for every v ∈ X n \ {u}, act as the identity on C(v). Given g ∈ G, there isg ∈ rist G (u) such thatg(us) = ug(s) for any s ∈ X N . Clearly, the map g →g is an isomorphism from G to rist G (u). Fix h ∈ V \ {e} and take w ∈ X n and z ∈ X m such that w = z, n ≥ m and h(ws) = zs for any s ∈ X N . Furthermore, take v ∈ X n−m such that zv = w. Given s ∈ X N , we have that (1) {ghg −1 (wvs) :g ∈ rist G (zv)} = {zvg(s) : g ∈ G}.
Since G X N is faithful, it follows from (1) that |h G | = ∞, thus proving the claim. From [GTWZ21, Theorem 1.5], we obtain that G ∂ p G is free and from Theorem 5.2, we conclude that V ∂ p G is free.
Remark 5.5. In [LBMB18, Theorem 1.5], Le Boudec and Matte Bon showed that Thompson's group V is C * -simple, hence V ∂ F V is free. However, their proof is done by showing that V does not admit non-trivial amenable URS, not by exhibitting a concrete topologically free V -boundary. It seems as an interesting problem to determine whether V ∂ p G is strongly proximal, thus providing an alternative proof of C * -simplicity of V .
Remark 5.6. In [BKKO17, Theorem 1.4], it was shown that the class of C * -simple groups is closed by taking normal subgroups. Obviously, this class is not closed by taking commensurated subgroups, since any finite subgroup is commensurated. Moreover, Example 5.4 shows that, given Λ ≤ c Γ such that Γ is icc relatively to Λ, C * -simplicity of Γ does not pass to Λ in general.