Uniform symplicity of groups with proximal action

We prove that groups acting boundedly and order-primitively on linear orders or acting extremly proximality on a Cantor set (the class including various Higman-Thomson groups and Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms) are uniformly simple. Explicit bounds are provided.


. Introduction
Let Γ be a group. It is called N-uniformly simple if for every nontrivial f ∈ Γ and nontrivial conjugacy class C ⊂ Γ the element f is the product of at most N elements from C ± . A group is uniformly simple if it is N-uniformly simple for some natural number N. Uniformly simple groups are called sometimes, by other authors, groups with finite covering number or boundedly simple groups (see e.g. [, , ]). We call Γ boundedly simple, if N is allowed to depend on C. The purpose of this paper is to prove results on uniform simplicity, in particular Theorems ., ., and . below, for a number of naturally occurring infinite permutation groups.
Every uniformly simple group is simple. It is known that many groups with geometric or combinatorial origin are simple. In this paper we prove that, in fact, many of them are uniformly simple.
Below are our main results.
Let (I, ≤) be a linearly ordered set. Let Aut(I, ≤) denote the group of order preserving bijections of I. We say that g ∈ Aut(I, ≤) is boundedly supported if there are a, b ∈ I such that g (x) x only if a < x < b. The subgroup of boundedly supported elements of Aut(I, ≤) will be denoted by B(I, ≤). Theorem . (Theorem . below). Assume that Γ < B(I, ≤) is proximal on a linearly ordered set (I, ≤) (i.e. for every a < b and c < d from I there exists g ∈ Γ such that g(a) < c < d < g (b)). Then its commutator group Γ ′ is six-uniformly simple and the commutator width of this group is at most two.
For the definition of a commutator width of a group see the beginning of Section .
Observe that every doubly-transitive (i.e. transitive on ordered pairs) action is proximal.
This theorem immediately applies e.g. to B (Q, ≤) and the class of Higman-Thomson groups F q,r , for q > r ≥ , where the latter are defined as follows. We fix natural numbers q > r ≥ . The Higman-Thompson group F q,r is defined as piecewise affine, order preserving transformations of ((, r) ∩ Z[  /q], ≤) whose breaking points (i.e. singularities) belong to Z[  /q] and the slopes are q k , for k ∈ Z (see [, Proposition .]). The Thompson group F is the group F , in the above series. Moreover, F q,r is independent of r (up to isomorphism) [, .]. The Higman-Thompson groups satisfy the assumptions of Theorem . due to Lemmata ., . from Section .
Our result implies that Γ = B(Q, ≤) is six-uniformly simple. Whereas Droste and Shortt proved in [, Theorem .(c)] that B(Q, ≤) is two-uniformly simple. In fact, they proved that if Γ < B(I, ≤) is proximal (they use the term 'feebly -transitive' for proximal action) and additionally closed under ω-patching of conjugate elements, then Γ is two-uniformly simple. Thus, our Theorem . covers larger class of examples than that from [] (as we assume only proximality), but with slightly worse bound for uniform simplicity.
The uniform simplicity of the Thompson group F = F , was proven implicitly by Bardakov, Tolstykh, and Vershinin [, Corollary .] and Burago and Ivanov []. Although their proofs generalise to the general result given above, we write it down for several reasons. Namely, in above cited papers some special properties of the linear structure of the real line is used, while the result is true for a general class of proximal actions on linearly ordered sets. The Droste and Shortt argument uses ωpatching, which is not suitable for our case. Furthermore, although in the examples mentioned above the action is doubly-transitive, the right assumption is proximality, which is strictly weaker than double-transitivity. In Theorem . we construct a bounded and proximal transitive action which is not doubly-transitive. This is discussed in details in Section . The second reason for proving Theorem . is that a topological analogue of proximality, namely extremal proximality (see the beginning of Section ), plays a crucial role in the proofs of the subsequent results. Extremal proximality was defined by S. Glasner in [, p. ] and [, p. ] for a general minimal action of a group on a compact Hausdorff space.
In Section  we go away from order preserving actions, and consider groups acting on a Cantor set, and also groups almost acting on trees. The following theorem is Corollary . ().
Theorem .. The commutator subgroup N ′ q of the Neretin group N q of spheromorphisms and the commutator subgroup G ′ q,r of the Higman-Thomson group G q,r are nineuniformely simple. The commutator width of each of those groups is at most three.
The group N q was introduced by Neretin in [, ., .] as the group of sphereomorphisms (also called almost automorphisms) of a (q + )-regular tree T q . We will recall the construction in Section .
The Higman-Thompson group G q,r is defined as the group of automorphisms of the Jónsson-Tarski algebra V q,r [, a.]. It can also be described as a certain group of homeomorphisms of a Cantor set [, p. ]. Moreover, one can view G q,r as a subgroup of G q, and the latter as a group acting sphereomorphically on the (q + )regular tree [, Section .], that is, they are subgroups of N q . If q is even, then [, Theorem .]. The group G , is also known as the Thompson group V. It is known that F q,r < G q,r .
Given a group Γ acting on a tree T, in the beginning of Section  we will define, following Neretin, the group Γ of partial actions on the boundary of T. Theorem . is a corollary of a more general theorem about uniform simplicity of partial actions.
Theorem .. Assume that a group Γ acts on a leafless tree T, whose boundary is a Cantor set, such that Γ does not fix any proper subtree (e.g. Γ\T is finite) nor a point in the boundary of T. Then the commutator subgroup Γ ′ of Γ is nine-uniformly simple. This is an immediate corollary of Theorems . and .. The latter theorem concerns with several characterisations of extremal proximality of the group action on the boundary of a tree. Section  is devoted to the proof that the groups of quasi-isometries and almostisometries of regular trees are five-uniformly simple.
The uniform simplicity of homeomorphism groups of certain spaces has been considered since the beginning of s e.g. by Anderson []. He proved that the group of all homeomorphisms of R n with compact support and the group of all homeomorphisms of a Cantor set are two-uniformly simple (and has commutator width one). His arguments uses an infinite iteration arbitrary close to every point, which is not suitable for the study of spheromorphisms group and the Higman-Thomson groups.
N-uniform simplicity is a first order logic property (for a fixed natural number N). That is, can be written as a formula in a first order logic. Therefore, N-uniform simplicity is preserved under elementary equivalence: if G is N-uniform simple, then all other groups, elementary equivalent with G are also N-uniformly simple. In particular, all ultraproducts of Neretin groups, and Higman-Thompson groups mentioned above, are nine-uniformly simple.
Another feature of uniformly simple groups comes from [, Theorem .], where the second author proves the following classification fact about actions of uniformly simple groups (called boundedly simple in []) on trees: if a uniformly simple group acts faithfully on a tree T without invariant proper subtree or invariant end, then essentially T is a bi-regular tree (see Section  for definitions).
In Section  we discuss the primitivity of actions on linearly ordered sets (i.e. lack of proper convex congruences). In fact, we prove that primitivity and proximality are equivalent notions for bounded actions (Theorem .). Our primitivity appears in the literature as o-primitivity, see e.g. [, Section ].
Calegari [] proved that subgroups of pl + homeomorphism of the interval, in particular the Thompson group F, have trivial stable commutator length. Essentially by [, Lemma .] by Burago, Polterovich, and Ivanov (that we will explain for the completeness of the presentation) we reprove in Lemma . the commutator width of the commutator subgroup (and other groups covered by Theorems . and .) of the Thompson group F. Let us discuss examples and nonexamples of bonded and uniform simplicity. It is known that a simple Chevalley group (that is, the group of points over an arbitrary infinite field K of a quasi-simple quasi-split connected reductive group) is uniformly simple [, ]. In fact, there exists a constant d (which is conjecturally , at least in the algebraically closed case []) such that, any such Chevalley group G is d · r(G)uniformly simple, where r(G) is the relative rank of G [].
Full automorphism groups of right-angled buildings are simple, but never boundedly simple, because of existence of nontrivial quasi-morphisms [], [, Theorem .] (except if the building is a bi-regular tree [, Theorem .]).
Compact groups are never uniformly simple. More generally, a topological group Γ is called a sin group if every neighborhood of the identity e ∈ Γ contains a neighborhood of e which is invariant under all inner automorphisms. Every compact group is sin (as if V is such a neighborhood, then γ∈Γ γ − Vγ has nonempty interior). Clearly every infinite Hausdorff sin-group is not uniformly simple. Moreover, many compact linear groups (e.g. SO (, R)) are boundedly simple, because of the presence of the dimension with good properties. (See also the discussion at the end of this section.) Let us conclude the introduction by relating simplicity and the notion of central norms on groups. Let Γ be a group. A function · : Γ → R ≥ is called a seminorm if A seminorm is called A seminorm is discrete if and only if the topology it induces is discrete. A discrete seminorm is a norm. Every central seminorm · is conjugacy invariant: ghg − = h .
A typical example of a nontrivial central and discrete norm is a word norm · S attached to a subset S ⊆ Γ (cf. [, .]): For a nontrivial central norm · we define an invariant ∆( · ) = sup g∈Γ g inf  g∈Γ g . Of course, if · is either nondiscrete or unbounded then ∆( · ) = ∞. We define ∆(Γ) to be the supremum of ∆( · ) for all nontrivial central norms on Γ.
Proposition .. Let Γ be a group. Then () Γ is simple if and only if any nontrivial central seminorm on Γ is a norm; () Γ is boundedly simple if and only if every central seminorm on Γ is a bounded norm; () If Γ is uniformly simple, then every central seminorm on Γ is a bounded and discrete norm; () Proof. () It is obvious that the kernel of a central seminorm is closed under multiplication and conjugacy invariant. Thus, it is a normal subgroup. On the other hand, if Γ  ⊳ Γ is a proper normal subgroup of Γ then () Suppose that · is a central seminorm and assume that Γ is boundedly simple. Choose  g ∈ Γ. There exists N = N(g) such that every element f is a product of at most N conjugates of g and g − . Thus, by the triangle inequality and conjugacy invariance, f ≤ N g . The number N g is finite and independent on f . For the converse take  g ∈ Γ and consider the word norm · S attached to S = Γ g ∪ Γ g − = {x ∈ Γ : x is conjugated to g or g − }. It is obvious that · S is a central seminorm on Γ.
Thus Γ is boundedly simple, as · S is bounded.
Suppose Γ is N-uniformly simple, i.e. N is independent on g ∈ Γ and take nontrivial central norm · . By the triangle inequality we conclude that ∆( · ) ≤ N, which proves the necessity of the condition in (). For the converse, take  g ∈ Γ, and consider the word norm · g above. We have that Γ is ∆( · g ) ≤ ∆(Γ)-uniformly simple. This completes the proof of (), which implies ().
In particular, the infinite alternating group A ∞ = n≥ A n is simple but not boundedly simple. To see the later, observe that the cardinality of the support is an unbounded central norm. This norm is maximal up to scaling. Indeed, essentially by [, Lemma .] every element σ ∈ A ∞ is a product of at most # supp(π) conjugates of π. Also, it is easy to see that SO() is boundedly simple, but not uniformly simple. The angle of rotation is a full invariant of an element of that group, and this function is a central norm. Clearly it is not discrete. As before, one can observe that if R is a rotation by an angle θ, then every other rotation can be obtained by at most π θ conjugates of R.
Every universal sofic group [, Section ] is boundedly simple, but not uniformly simple. Namely, let (S n , · H ) n∈N be the full symmetric group on n letters with the normalized Hamming norm: σ H =  n | supp(σ)|, for σ ∈ S n . Take the metric ultraproduct of (S n , · H ) n∈N over a nonprincipal ultrafilter U . Then, the proof of [, Proposition .()] shows that G is boundedly simple. Furthermore, G is not uniformly simple, as · is a non-discrete central norm on G (see Proposition .()). As before, this norm is maximal up to scaling due to [, Lemma .].

. Burago-Ivanov-Polterovich method
The symbol Γ will always denote a group. For a, b ∈ Γ we use the following notation: By Γ g we mean the conjugacy class of g ∈ Γ.
Let C be a nontrivial conjugacy class in Γ. By C-commutator we mean an element of If h ∈ C we will use the name h-commutator as a synonym of C-commutator, for short. Of course [Γ, C] = C.C − , thus the set of C-commutators is closed under inverses and conjugation.
The commutator length of an element g ∈ [Γ, Γ] is the minimal number of commutators sufficient to express g as their product. The commutator width of Γ is the maximum of the commutator lengths of elements of its derived subgroup [Γ, Γ] = Γ ′ .
We say that f and g ∈ Γ commute up to conjugation if there exist h ∈ Γ such that f and h g commute.
Lemma .. Assume that α and h β commute. Then [α, β] is a product of two h-commutators. More precisely [α, β] can be written as a product of two conjugates of h and two conjugates of h − by elements from the group generated by α and β.
. In particular, elements of a displaceable subgroup commute up to conjugation.
Burago, Polterovich, and Ivanov [, Theorem .(i)] proved that if for every k ∈ N some conjugate of g k-displaces H then every element of H ′ is a product of seven g-commutators. We get a better result under a stronger assumption.
Proposition .. Assume that g ∈ Γ is such that for every finitely generated subgroup H < Γ and k ∈ N, there exists a conjugate of g which k-displaces H. Then every element of Γ ′ is a product of two commutators in Γ and three g-commutators in Γ. Moreover, Proof. Every element f ∈ Γ ′ can be expressed as a product of k commutators of k elements of Γ, for some k ∈ N. Call the group they generate H. Since some conjugate Since some conjugate of g displaces the group generated by α and β, by Lemma ., [α, β] is a product of two g-commutators. Thus f is a product of three g-commutators.
The moreover part follows from the fact that [γ, h] ∈ Γ g Γ g − and that Γ h and Γ ′ h are equal. The last claim follows from the fact that if f ∈ Γ commutes up to comutation with g f − commute. Then Note that the assumption of the above corollary implies that neither Γ nor Γ ′ are finitely generated. However, we will use this approach to prove uniform simplicity of the Higman-Thompson groups which are known to be finitely generated.
Lemma .. Assume that every two elements in Γ commute up to conjugation. Then every commutator in Γ can be expressed as a commutator in Γ ′ . In particular, Γ ′ = Γ ′′ is perfect.
Proof. Let α and β belong to Γ. Choose h and g such that α and h β commute and also Proof. By Lemma . every element of Γ ′  is a product of two commutators of Γ  . By Lemma . they can be chosen to be commutators of elements of Γ ′  . By Lemma . each of them is a product of two g-commutators over Γ ′  .

. Bounded actions on ordered sets
The purpose of this section is to prove that numerous simple Higman-Thompson groups acting as order preserving piecewise-linear transformations are, in fact, uniformly simple.
We always assume that a group Γ acts faithfully on the left by order preserving transformations on a linearly ordered set (I, ≤). Given a map g : I → I, we define the support supp(g) of g to be {x ∈ I : g(x) x}. Given a and b ∈ I we define (a, b) = {y ∈ I : a < y < b}. By (a, ∞) we will denote the set {x ∈ I : a < x}. The group of all bounded automorphisms of (I, ≤) is denoted by B(I, ≤).
We call such an action Note, that being proximal implies that (I, ≤) is dense without endpoints.
Theorem .. Assume that Γ acts faithfully, order preserving, boundedly, and proximally on a linearly ordered set (I, ≤). Then its commutator group Γ ′ is six-uniformly simple and the commutator width of Γ ′ is at most two.
Proof. We apply Proposition .. Let g be an arbitrary nontrivial element of Γ ′ . Let a ∈ I be such that g(a) a. Replacing g by g − we may assume that a < g(a). Choose b ∈ I such that a < b < g(a). Then g(a, b) ∩ (a, b) = ∅. Let H be an arbitrary finitely generated subgroup of Γ. Then, there exists an interval, say (c, d), containing supports of all generators of H, hence also containing supports of all elements of H. By the proximality of the action, we may assume (possibly conjugating g), that (c, d) ⊆ (a, b). It is clear that such conjugate of g ∞-displaces H. Thus Proposition . applies.
Let us apply Theorem . to the Higman-Thompson groups of order preserving piecewise-linear maps. We first recall the definitions. Let q > r ≥  be integers. Recall that F q,r (F q respectively) is defined as piecewise affine (we allow only finitely many pieces), order preserving bijections of ((, r) ∩ Z[  /q], ≤) ((Z[  /q], ≤) respectively) whose breaking points of the derivatives belong to Z[  /q] and the slopes are q k , for k ∈ Z (see the bottom of page  and the top of page  in []).
Define BF q,r (BF q respectively) to be the subgroup of F q,r (F q respectively) consisting of all such transformations γ that are boundedly supported, that is, supp(γ) ⊆ (x, y), We use the following lemma below. The first part of it is a known result [].
The groups BF q,r and BF q are isomorphic ( [, Proposition c.]). () The commutator subgroups of F q,r and BF q,r are equal.
Proof. () It is obvious that BF ′ q,r ⊆ F ′ q,r . Let us prove ⊇. Note that F ′ q,r ⊆ BF q,r (because for g  , g  ∈ F q,r , the element [g  , g  ] acts as the identity in some small neighbourhoods of  and r).
We consider the action of BF q on Z[  /q] and its orbits.
BF q acts in a doubly-transitive way on I. In particular, the action is proximal.
As a corollary of the above lemmata we get that groups F q,r satisfy the assumptions of Theorem ..
BF ′ q is six-uniformly simple and the commutator width of it is at most two.
Remark .. Theorem . applies to the following groups.
• Bieri and Strebel [] define more general class of groups acting boundedly on R. They take a subgroup P in the multiplicative group R > and a Z[P]-submodule A < R and define Γ := B(R; A, P) to be a group of boundedly supported automorphisms of R consisting of piecewise affine maps with slopes in P and singularities in A. They define an augmentation ideal I = p − |p ∈ P of Z[P] and prove that Γ acts highly transitive on IA. Thus Γ ′ is six-uniformly simple. • Another example of doubly-transitive and bounded action on a linear order (thus satisfying the assumptions of Theorem .) was considered by Chehata in [], who studied partially affine transformations of an ordered field and proved that this group is simple. Theorem . implies that the Chehata group is six-uniformly simple.

. Proximality, primitivity, and double-transitivity
In this section we prove (Theorem .) that proximality (from the previous Section) and order-primitivity are equivalent properties for bounded group actions. In general, these properties are inequivalent. The action of the group of integers on itself is primitive but neither proximal nor bounded. We also give an example of bounded, transitive and proximal action, which is not doubly-transitive (Theorem .).
Theorem .. Every proximal action is primitive. Any bounded and primitive action is proximal.
Proof. Assume the action is not primitive. Choose a, b and d such that a b and ψ(a) = ψ(b) ψ(d). Reversing the order if necessary, we may assume ψ(b) < ψ(d). Set c = a. This choice contradicts proximality, as if g(b, a) ⊆ (d, c) then Assume that action is bounded, but not proximal. Let a, b, c, and d witness the latter. For x, y ∈ I, x < y consider the relation ∼ x,y on I defined as It is enough to prove that ≃ c,d is not total, that is, e c,d f for some e, f ∈ I, because then the quotient map proves nonprimitivity of the action (I/ ≃ c,d has a natural Γ-action).
First, we claim that there is γ ∈ Γ such that γ(c) ≥ d. Indeed, if there is no such group element, define a map ψ : I → {, } by the formula This map would contradict primitivity.
Choose e and f from I such that supp(γ) ⊆ (e, f ). Then γ t (c, d) : t ∈ Z is a countable family of intervals in (e, f ), which are pairwise disjoint. We claim that e c,d f , as otherwise there are x, y ∈ [e, f ], x < y such that x ∼ c,d y and (x, y) contains γ t (c, d) for some t ∈ Z, which is impossible.
Clearly, if Γ acts proximally on (I, ≤), then in acts in a such way on any orbit. Thus, we will restrict to transitive actions.
Examples of actions we discuss above are doubly-transitive (cf. Lemma . () and Remark .). Thus they are proximal. This property seems to be easier to check than doubly-transitivity. We construct below an example of bounded, transitive and proximal action, which is not double-transitive. The reader may consult this result with a result of Holland [, Theorem ], which says that every bounded, transitive, primitive and closed under min, max action must be doubly-transitive. Moreover, any group acting boundedly and transitively cannot be finitely generated. Indeed, finite number of elements have supports in a common bounded interval, thus the whole group is supported in that interval, so does not act transitively.
Proof. For each k ∈ N we will define a countable linear order (I k , ≤), a group Γ k acting on it, and a function f k : I k × I k → Z such that: ) for k > , Γ k acts transitively and proximally on I k by order preserving transformations (but not doubly-transitive); Then we take Γ ∞ = k∈N Γ k , which acts boundedly, transitive and proximally, but not doubly-transitive on Since (I ∞ , ≤) is a countable and, by proximality, dense linear order without ends, it is isomorphic to (Q, ≤).
In the following inductive construction we will define three auxiliary points i − k < i k < i + k from I k . We put Γ  = Z and I  = Z, where Γ  acts on I  by translations. Let f  (n, m) = n − m and Assume we have constructed I k , Γ k , and f k . Let I k+ = a ∈ I k Z : ∀ ∞ n ∈ Z a(n) = i k and i k+ (n) = i k for all n ∈ Z. In plain words, I k+ consists of all functions from Z to I k which differ from a constant function (denoted by i k+ ) taking the value i k , only at finite many places. Define a linear order on I k+ by putting a < b if min {n ∈ Z : a(n) < b(n)} < min {n ∈ Z : a(n) < b(n)}, with the convention that min ∅ > n for all n ∈ Z. Note that I k embeds into I k+ : Consider Conv(I k ) = {a ∈ I k+ : a(n) = i k for all n < }, with the following action of Γ k : Extend the action of Γ k to the whole of I k+ by the identity on the complement I k+ Conv(I k ). Thus the action of Γ k on I k+ is bounded. Define yet another automorphism σ k+ of I k+ by (σ k+ a)(n) = a(n + ). Let Γ k+ to be the group generated by Γ k and σ k+ . The action of Γ k+ on I k+ is clearly transitive.
For every pair a b from I k+ , define m a,b = min{n ∈ Z : a(n) b(n)}.
For a < b and c < d let γ ∈ Γ k be such that (c(m c,d ), d(m c,d )) ⊆ γ (a(m a,b ), b(m a,b )) (such γ exists by proximality of the action of Γ k on I k ).  (a, b), which proves the proximality of the action of Γ k+ on I k+ .
Finally, define f k+ (a, b) = f k (a(m a,b ), b(m a,b )). Clearly, f k+ is Γ k+ -invariant, hence the action of Γ k+ on I k+ is not doubly-transitive.
The element σ k ∈ Γ k stabilizes i k and has unbounded orbits on (i k , ∞) ⊂ I k . Thus the stabiliser of i ∞ = lim i k has unbounded orbits on (i ∞ , ∞) ⊂ I ∞ . This is enough to conclude that the action is proximal.
Question .. Is there any transitive, proximal bounded action without the property that point stabilisers have unbounded orbits?

. Extremely proximal actions on a Cantor set and uniform simplicity
The main goal of the present section is prove Theorem ., which gives a criterion for a group acting on a Cantor set to be nine-uniformly simple.
Let C be a Cantor set. Assume that a discrete group Γ acts on C by homeomorphisms. By the topological full group Γ < Homeo(C) of Γ we define (see e.g. []) Γ = g ∈ Homeo(C) : for each x ∈ C there exists a neighbourhood U of x and γ ∈ Γ such that g| U = γ| U .
Through this section we assume that: • group Γ acts faithfully by homeomorphisms on a Cantor set C; • Γ is a topological full group, i.e. Γ = Γ ; • the action is extremely proximal, i.e. for any nonempty and proper clopen sets The second assumption is not hard to satisfy as Γ = Γ .
Theorem .. Assume that Γ satisfies the above assumptions. Then Γ ′ , the commutator subgroup of Γ, is nine-uniformly simple. The commutator width of Γ ′ is at most three. Therefore, if Γ is perfect (i.e. Γ ′ = Γ), then Γ is nine-uniformly simple.
Before proving ., we need a couple of auxiliary lemmata.
Suppose x ∈ C and h ∈ Γ. By the Hausdorff property of C, if h(x) x, then there exists a clopen subset U ⊂ C containing x, such that h(U) ∩ U = ∅. In such a situation we define an element τ h,U ∈ Γ exchanging U and h(U): Such an element belongs to Γ, since Γ = Γ is a topological full group. Observe that τ  h,U = id and f τ h,U f − = τf h,f (U) , for f ∈ Γ. Lemma .. Assume Γ acts extremely proximally on a Cantor set C.

() Γ ′ acts extremely proximally on C. () For any nontrivial f ∈ Γ and a proper clopen V C there is
() Let f , g ∈ Γ be nontrivial. Then there is h ∈ Γ ′ such that h g.f is supported outside a clopen subset.
Proof. () Let U and V be nonempty and proper clopen subsets of C. Shrinking U, if necessary, we may assume that U ∪ V C (that is, we may always take g ∈ Γ and It is straightforward to check that, since U, g  (U), and g  (U) are pairwise disjoint, we have g  =  which is equivalent to And similarly for h. In particular, g and h belong to Γ ′ . Furthermore, ()). If h  gf is the identity on U the proof is finished. Otherwise define γ = h  g. We may find W ⊂ U such that γf (W)∩W = ∅ and W∪f (W)∪γ − (W) C. Notice that γ − (W), W and f (W) are pairwise disjoint.
For any clopen U ⊂ C, let Γ U be the subgroup of Γ consisting of elements of Γ supported on U.

Lemma .. Let V C be a proper clopen set. Then there exists a proper clopen
Then ψ is a homeomorphism, which induces an isomorphism Ψ : Γ → Γ U given by for any h ∈ Γ and x ∈ C. Since Ψ is the identity on Γ V , Ψ(f ) = f , for any f ∈ Γ V . Therefore, if f ∈ Γ ′ , then f ∈ Γ ′ U . Lemma .. Assume that U V ⊆ C are clopens. There exists h ∈ Γ ′ V such that for all k ∈ Z, the sets h k (U) are pairwise disjoint.
Proof of Theorem .. Let f be an element of Γ ′ and A be a nontrivial conjugacy class of Γ ′ . By Lemmata . () and We claim that f  is a product of four A-commutators in Γ ′ . Choose V  V  C and ω ∈ V  V  . We apply Proposition .. Namely, let Γ  denote the union of groups Γ V , such that V is a clopen contained in V  {ω}. Clearly, Γ  is a proper subgroup of Γ V  . By Lemma .(), we may choose g ∈ A, such that g(V  )∩ V  = ∅. Thus, g displaces Γ  . Let H be a finitely generated subgroup of Γ  . The union of supports of its generators is a clopen U, properly contained in V  , since ω U.
Then h ∞-displaces H. Thus Proposition . applies and f  ∈ Γ ′ V  < Γ ′  is a product of four g-commutators. By Lemma ., the commutator width of Γ ′  is at most two. By Lemma . (), every element decomposes as a product of a conjugate of a given nontrivial element from Γ ′ , say a commutator, and an element conjugate into Γ ′  . Thus every element of Γ ′ is a product of three commutators.

. Groups almost acting on trees
In this section we apply Theorem . to groups almost acting on trees.
By a graph (whose elements are called vertices) we mean a set, equipped with a symmetric relation called adjacency. A path is a sequence of vertices indexed either by a set {, . . . , n} or N (in such a case we call the path a ray) such that consecutive vertices are adjacent, and no vertices whose indices differ by two coincide (i.e. there are no backtracks). A graph is called a tree if is connected (nonempty) and has no cycles, i.e. paths of positive length starting and ending at the same vertex (in particular, the adjacency relation is irreflexive).
Ends of T are classes of infinite rays in T. Two rays are equivalent if they coincide except for some finite (not necessarily of the same cardinality) subsets. The set of all ends of T is denoted by ∂T, and is called the boundary of T.
Given a pair of adjacent vertices (called an oriented edge) e = (v, w), we call the set of terminal vertices of paths starting at e a halftree of T and we will denote it by T e . The classes of rays starting at e will be called the end of a halftree T e and will be denoted by ∂T e ⊂ ∂T. By − e we denote the pair (w, v).
We endow ∂T with a topology, where the basis of open sets consist of ends of all halftrees.
A valency of a vertex v is the cardinality of the set of vertices adjacent to v. A vertex of valency one is called a leaf. If every vertex has valency at least three but finite, then the boundary ∂T is easily seen to be compact, totally disconnected, without isolated points, and metrizable. Thus, ∂T is a Cantor set. In such a case, every end ∂T e of a halftree is a clopen (open and closed) subset of ∂T.
A spheromorphism is a class of permutations of T which preserve all but finitely many adjacency (and nonadjacency) relations. Two such maps are equivalent if they differ on a finite set of vertices (see e.g. [, Section ]). We denote the group of all spheromorphisms of T by AAut(T). If T is infinite, then the natural map Aut(T) → AAut(T) is an embedding. Every sphereomorphism f ∈ AAut(T) induces a homeomorphism of its boundary ∂T.
For an integer q > , by T q we denote the regular tree whose vertices have degree (q+). The group N q was introduced by Neretin in [, ., .] as the group AAut(T q ) of sphereomorphisms of (q + )-regular tree T q . It is abstractly simple [].
In what follows, we will be interested in subgroups Γ < Aut(T) acting extremely proximally on the boundary ∂T (see Theorem . and Corollary . below). The whole group of automorphisms Γ = Aut(T q ) of T q is such an example. Another example (cf. Example .) is the automorphism group Γ = Aut(T s,t ) of a bi-regular tree T s,t , s, t >  (i.e. every vertex of T s,t is black or white, every black vertex is adjacent with s white vertices, every white -with t black vertices). We prove that the group Γ of partial Γ-actions on ∂T is then nine-uniformly simple.
The group Aut(T s,t ) itself is virtually -uniformly simple [, Theorem .]. (Bounded simplicity in [] means uniform simplicity in our context.) There is a connection between the notion of a sphereomorphism and topological full group acting on a boundary of a tree.
() Any subdivision of ∂T into clopens can be refined to U  , a subdivision into ends of halftrees (since any clopen in ∂T is a finite union of boundaries of halftrees). Therefore the Neretin group N q can be characterized as N q = Aut(T q ) = AAut(T q ).
() Another, well studied, example comes from considering Aut  (T q ) = automorphisms of T q preserving chosen cyclic orders on edges adjacent to any vertex of T q .
One may induce cyclic orders by planar representation of T q . The group Aut  (T q ) is the Higman-Thompson group G q, [, Section ], [, .]. () Those two examples can be generalized in the following manner (see [, Section .]).
Let c : E(T q ) → {, . . . , q} be a function from the set E(T q ) of (undirected) edges of (q + )-regular tree T q , such that for every vertex v, the restriction of c to the set of edges E(v) starting at v gives a bijection with {, . . . , q}. We say that such c is a proper colouring of T q . Let F < S q+ be a subgroup of permutations of {, . . . , q}. Using proper colouring c and F we define the universal group U(F) to be In fact U(F) is independent (up to conjugation is Aut(T q )) of the choice of proper colouring c. We prove (see Corollary .) that U(F) ′ is nine-uniformly simple, provided that F is transitive on {, . . . , q}. If F is generated by a (q + )-cycle, then U(F) = Aut  (T q ), from (). If F = S q+ , then U(F) = Aut(T q ).
We call an action for a group Γ on a tree T minimal if there is no proper Γ-invariant subtree of T. Given a subset A of a tree. We define its convex hull to be the set of all vertices which lie on paths with both ends in the set A. It is a subtree. The action is minimal if and only if the convex hull of any orbit is the whole tree.
Example .. Every action on a leafless tree with a finite quotient is minimal. The converse is not true (see Example .).
Indeed, the distance from a Γ-orbit is a bounded function. Hence the complement of an orbit cannot contain an infinite ray. Thus every vertex lies on a path with endpoints in a given orbit. Lemma . ([, Lemma .]). Assume that a group Γ acts minimally on a leafless tree T. Then for every vertex v and an edge e the orbit Γv intersects the halftree T e .
Proof. If Γv is all contained in T − e , so is its convex hull. Thus the claim.
We call an action for a group Γ on a tree T parabolic if Γ has a fixed point in ∂T.
An action of a group by homeomorphisms on a topological space is called minimal if there is no proper nonempty closed invariant set (equivalently, if every orbit is dense). This notion should not cause confusion with the notion of minimal actions on trees. (A tree is a set equipped with a relation as opposed to its geometric realisation which is a topological space.) Theorem .. Assume that T is a leafless tree such that ∂T is a Cantor set. Let Γ act on T.
The following are equivalent. () The action of Γ on ∂T is extremely proximal (see the beginning of Section  for the definitions). () The action of Γ on ∂T is extremely proximal. () The action of Γ on ∂T is minimal and ∂T does not support any Γ-invariant probability measure. () The action of Γ on T is minimal and not parabolic, that is, there is no proper Γ-invariant subtree of T and Γ has no fixed point in ∂T.
( ⇒ ) Let F be a closed, nonempty, proper and Γ-invariant subset of ∂T. Choose x ∈ F and a proper clopen V ⊂ ∂T containing x. Define U = ∂T F. Then, there is no g ∈ Γ such that g(V) ⊂ U, since g(x) = γ(x) ∈ F, for some γ ∈ Γ; thus contradiction.
Similarly, let µ be a Γ-invariant measure on ∂T.
where U i 's are disjoint nonempty clopens. We may assume that µ( is a contradiction. Hence, the action is not extremely proximal.
Suppose that there exists a finite Γ-invariant subtree T ′ of T. We use the following definition. Given a vertex v of T, we define the visual measure associated to v to be the unique measure µ v on ∂T with the following property: if {v i } n i= is any injective path starting at v  = v, then where d i is the valence of v i . The visual metric µ v is obviously invariant under the action of the stabiliser Stab(v) of v in Aut(T).
We can consider the average of the visual measures associated to the vertices of this subtree T ′ . It will be a Γ-invariant measure on ∂T.
( ⇒ ) By Lemma . we may assume that, for every pair of edges e and f , there is γ ∈ Γ such that either T γ e or T −γ e is strictly contained in T f . It is enough to show that one can find γ ∈ Γ, such that the later holds, i.e. ∂T −γ( e) ∂T f (indeed, since ends of halftrees constitute a basis, we can find edges e and f such that ∂T e ⊂ U and ∂T f ⊂ C V, for nonempty proper clopens V and U in ∂T; if there is γ ∈ Γ such that ∂T −γ( e) ∂T f then γV ⊆ ∂T −γ( e) ∂T f ⊆ U).
It is enough to prove this claim for e = f . Indeed, if there exists γ  ∈ Γ such that T γ  e T f and T −γ  e T e , then T −γ  γ  e T γ  e T f .
Assume that there exists γ ∈ Γ such that T γ e T e . Let {v i } n i= be a path such that e = (v  , v  ) and γ e = (v n− , v n ). Then {v i } i∈Z , defined as v nq+r = γ q v r , is a biinfinite path. Let ω be its end as i → ∞. Choose η ∈ Γ such that η(ω) ω. Consider the biinfinite path from ω to η(ω). It coincides with {v i } i<i − and {ηv −i } i>i + for some i ± ∈ Z. Therefore T −ηγ k e T γ k e , for k big enough. Hence, T −γ −k ηγ k e T e . Thus the claim.
Remark .. Only clause () from . concerns an action of a group on a tree. The other parts of . are about actions on a Cantor set. We do not know if there is a straight argument for proving equivalence of () and () from Theorem ., without referring to actions on trees.
Below is an application of Theorems . and . to the Neretin groups and the Higman-Thompson groups.
() Suppose F < S q+ is a transitive permutation subgroup and let c be a proper colouring of T q (see Example .()). Then U(F) acts transitively on the directed edges of T q , thus U(F) ′ is nine-uniformly simple. () Fix natural numbers q > r ≥ . The commutator subgroup N ′ q of the Neretin group N q , and the Higman-Thompson group G ′ q,r , are nine-uniformly simple and have commutator width bounded by three.
Proof. Let Γ = U(F). Then the action of Γ on T q is not parabolic as there is no Stab(v)fixed edge adjacent to v, hence no Stab(v)-fixed ray. It is minimal since the action is transitive.
Therefore, in case of the Neretin group N ′ q and the Higman-Thompson group G ′ q, , Theorem . applies immediately due to Theorem ..
Suppose F is a family of pairwise disjoint ends of halftrees ∂T e i ⊂ ∂T q , for  ≤ i ≤ q−r. If Γ F is a pointwise stabiliser of F in Aut  (T q ) (see Example .()), then Γ F is isomorphic to G q,r [, Section ]. Moreover, Γ F is its own topological full group acting extremely proximally on C = ∂T q q−r i= ∂T e i . Hence we get the conclusion for G ′ q,r . Corollary .. Suppose Γ = F n is a free group of rank n ≥ . Then Γ acts on its Cayley graph, which is T n− . This action is transitive and clearly not parabolic. Thus the induced action on the boundary is extremely proximal. Therefore F n ′ is nine-uniformly simple by Theorem .. [, p. ]). We apply our results to trees constructed by Tits. Any connected graph (G, E) of finite valence, with at least one edge, can appear as a quotient of a (finite valence) tree.
Assume that c is a function from oriented edges of G into the set of positive integers. By a result of Tits, there is a tree T and a group Γ acting on T such that G = Γ\T and, for any v, and w ∈ T such that (Γv, Γw) is an edge in G, there are exactly c(Γv, Γw) vertices in Γw adjacent to v (or none if it is not an edge of G).
If c is such that the sum over edges starting at a given vertex is at least three (but finite), then the boundary of T is a Cantor set.
If values of c are at least two, the group action of Γ on T is minimal and not parabolic [, .], i.e. the action of Γ on ∂T is extremely proximal due to Theorem ., and Γ is nine-uniformly simple due to Theorem ..
Corollary .. The groups of quasi-isometries and almost-isometries of a regular tree T q are five-uniformly simple.
Proof. This follows from Lazarovich results from the appendix. Let Γ be one of those groups. By Theorem . Γ = Γ ′ . Since Aut(T q ) is a subgroup of Γ, it acts extremely proximally on ∂T q (see Lemma .) as a topological full group (see Lemma .). This already proves nine-uniform simplicity.
Let  g and f be two elements of Γ. By Lemma . there exists g  , a conjugate of g, such that f  = g −  f fixes a clopen in ∂T q . By Lemma ., f  is a commutator of two elements fixing an open set in ∂T q . Thus, by Lemma ., f  is a product of two g-commutators.
. Appendix by Nir Lazarovich: Simplicity of AI(T q ) and QI(T q ) We begin by recalling the following definitions.
For λ ≥ , and K ≥ , a (λ, K)-quasi-isometry between two metric spaces (X, d X ) and and for all y ∈ Y there exists x ∈ X such that d Y (y, f (x)) ≤ K.
A map f is a quasi-isometry (resp. almost-isometry) if there exist K and λ (resp. K) for which it is a (λ, K)-quasi-isometry (resp. K-almost-isometry).
Two quasi-isometries f  , f  : X → Y are equivalent if they are at bounded distance (with respect to the supremum metric).
The group of all quasi-isometries (resp. almost-isometries) from a metric space X to itself, up to equivalence, is denoted by QI(X) (resp. AI(X)). Thus, for q ≥ , we have the following containments: Aut(T q ) ⊂ N q ⊂ AI(T q ) ⊂ QI(T q ) ⊂ Homeo(∂T q ).
Where the last containment follows from the following lemma.
Lemma .. The group QI(T q ) acts faithfully on ∂T q .
Proof. Let g ∈ QI(T q ) be a quasi-isometry. Let v ∈ T q , and let x  , x  , x  ∈ ∂T q be three distinct points such that v is the median of x  , x  , x  , that is, v is the unique intersection of all three (biinfinite) geodesics x  x  , x  x  , x  x  . Then, by the stability of quasi-geodesics in Gromov hyperbolic spaces [, Theorem .], gv is at bounded distance (which does not depend on the vertex v) from the midpoint of gx  , gx  , gx  . This implies that if g induces the identity map at the boundary, then g ∼ id.
In fact, the proof above is valid whenever the space X is a proper geodesic Gromov hyperbolic space X which has a Gromov boundary of cardinality at least three Światosław R. Gal and Jakub Gismatullin whose convex hull is at bounded distance from X (e.g. any non-elementary hyperbolic group).
For what follows, let Γ be the group QI(T q ) or AI(T q ) for q ≥ .
Lemma .. The group Γ < Homeo(∂T q ) is a topological full group.
Proof. Fix g ∈ Γ , and let ∂T e  , . . . , ∂T e n be a disjoint cover of ∂T such that g| ∂T e i = γ i | ∂T e i for some γ i ∈ Γ. For each  ≤ i ≤ n let e i, , . . . , e i,m be such that ∂T e i, , . . . , ∂T e i,m is a disjoint cover of g T e i . We may assume, by changing each γ i on a bounded set, that γ i (T e i ) = m j= T e i,j . Let us define for v ∈ T e i , and v otherwise. It is clear that if γ is in Γ, then it induces the element g on the boundary.
Let λ, K be the maximal quasi-isometry constants of γ i , and let M be the diameter of the bounded set { e  , . . . , e n , γ e  , . . . , γ e n }.
We claim the following: for all v, w ∈ T q , d(γv, γw) ≤ λd(v, w) + (K + M). Indeed, if v, w are both in some T e i or in T q \ n i= T e i , then the inequality is obvious. If v ∈ T e i and w ∈ T e j for some i j, then d(v, w) = d(v, e i ) + d(e i , e j ) + d(e j , w) and therefore d(γx, γy) =d(γ i x, γ i e i ) + d(γ i e i , γ j e j ) + d(γ j e j , γ j w) ≤λd(γv, γe i ) + K + M + λd(e j , w) + K ≤λd(x, y) + K + M.
Similarly, one shows this inequality for v ∈ T e i and w ∈ T q \ n i= T e i . Furthermore, the element γ ′ , defined as satisfies that for all v, w ∈ T q , d(γ ′ v, γ ′ w) ≤ λ ′ d(v, w) + (K ′ + M ′ ), for the appropriate λ ′ , K ′ , and M ′ . Moreover, it is easy to see that γγ ′ ∼ id ∼ γ ′ γ, from which we deduce that γ is a quasi-isometry.
Lemma .. Every element g in Γ that fixes an open set at the boundary is a commutator of two elements fixing a common set at the boundary.
Proof. Let supp g ⊂ T e . Let {x n } n∈Z be a biinfinite line geodesic contained in T − e and such that x  is the starting point of e.
Let t ∈ Aut(T q ) be a translation along {x n } n∈Z , and let f be the function defined by f (v) = t n g − (v) for v ∈ t n (T e ) and n ≥ , and v elsewhere.
The function f is in Γ since all the functions t n g − have the same quasi-isometry constants and [t, f ] = t f f − = g.
Let s be a -almost-isometry defined as otherwise.
Then we still have [ts, f ] = g as s commutes with f . However both st and f fix t − T e . Thus the claim.
Theorem .. The group Γ is perfect and has commutator width at most .
Proof. It suffices to show that each element of Γ can be written as a product of two elements of Γ which fix an open set at the boundary, as both of them are single commutators by Lemma ..
Let  g ∈ Γ, there exists ω ∈ ∂T such that g(ω) ω. Let T e be a halftree whose boundary contains ω and for which g∂T e and ∂T e are disjoint, and do not cover the whole of ∂T. Let h ∈ Γ = Γ be the map defined by: otherwise.
We see that hg fixes T e , and thus the claim.
Remark .. Since, for all q  , q  ≥ , the trees T q  and T q  are quasi-isometric, the groups QI(T q  ) and QI(T q  ) are isomorphic.