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Limits of contraction groups and the Tits core

The Tits core G^+ of a totally disconnected locally compact group G is defined as the abstract subgroup generated by the closures of the contraction groups of all its elements. We show that a dense subgroup is normalised by the Tits core if and only if it contains it. It follows that every dense subnormal subgroup contains the Tits core. In particular, if G is topologically simple, then the Tits core is abstractly simple, and if G^+ is non-trivial then it is the unique minimal dense normal subgroup. The proofs are based on the fact, of independent interest, that the map which associates to an element the closure of its contraction group is continuous.


Introduction
In his seminal paper [5], J. Tits uses his notion of BN-pairs to show the following result: if G is a simple algebraic group over a field k, then every subgroup of G(k) normalised by G(k) † either is central, or contains G(k) † . The group G(k) † is defined to be the subgroup of G(k) generated by the unipotent radicals of the k-defined parabolic subgroups. Tits deduces in particular that the quotient of G(k) † by its centre is simple as an abstract group.
In the present paper, we associate a subgroup G † to an arbitrary locally compact group G and call it the Tits core of G. It is defined as the subgroup of G generated by the closures of the contraction groups of all elements of G. Recall that the contraction group of an element g ∈ G is given by con(g) = {u ∈ G | g n ug −n → 1 as n → +∞}.
Thus by definition we have The Tits core is clearly normal, and even characteristic, in G, but it need not be closed a priori.
In case G is the group of k-rational points of a simple algebraic group G over a local field k, the contraction group of each element coincides with the unipotent radical of some k-defined parabolic subgroup, and is in particular closed (see [4,Lemma 2.4]). It follows that G † = G(k) † in this case; this justifies our choice of terminology for G † .
The main results of this paper may be viewed as analogues of the aforementioned theorem by Tits in the case of totally disconnected locally compact groups. For the sake of brevity, we shall write t.d.l.c. for totally disconnected locally compact. Theorem 1.1. Let G be a t.d.l.c. group and let D be a dense subgroup of G. If G † normalises D, then G † ≤ D. Corollary 1.3. Let G be a topologically simple t.d.l.c. group. If the Tits core G † is non-trivial, then it is the unique minimal dense subnormal subgroup of G.
As in the aforementioned situation studied by Tits, we obtain a result of abstract simplicity, which is immediate from Corollary 1.3. Corollary 1.4. Let G be a t.d.l.c. group. If G is topologically simple, then the Tits core G † is abstractly simple.
A t.d.l.c. group G with trivial Tits core will be called anisotropic; equivalently G is anisotropic if all its elements have trivial contraction group. This is certainly the case if G is discrete, or compact, or if every element of G is contained in a compact subgroup. It should be emphasised that non-discrete topologically simple groups may also be anisotropic: indeed, the examples of topologically simple t.d.l.c. groups constructed in [9, §3] are all unions of ascending chains of compact open subgroups, hence anisotropic. Notice however that such groups are not compactly generated; it is in fact an interesting open problem to determine whether a noncompact compactly generated t.d.l.c. group without infinite discrete quotient can be anisotropic.
In any case, the Tits core is contained in the kernel of all anisotropic quotients of G. More precisely, results by Baumgartner-Willis [1] imply the following characterisation.
Proposition 1.5. Let G be a t.d.l.c. group and N be a closed normal subgroup. Then G/N is anisotropic if and only if G † ≤ N. In particular G † is the unique smallest closed normal subgroup of G affording an anisotropic quotient.
Although the results above highlight an analogy between the Tits core of a t.d.l.c. group and that of a simple algebraic group, the methods of proofs are necessarily completely different (in particular a t.d.l.c. group need not have a BN-pair, even if it is topologically simple). The main tool in proving Theorem 1.1 is an analysis of the behaviour of the contraction groups associated to a converging sequence of elements in G. In order to describe our results in this direction, we recall that the nub of the element g in a t.d.l.c. group G is defined by nub(g) = con(g) ∩ con(g −1 ).
The nub is a compact subgroup normalised by g; it coincides with the unique maximal compact subgroup of G normalised by g and on which g acts ergodically (see [10] and Theorem 2.2 below). A key property of the nub is that nub(g) is trivial if and only if con(g) is closed (see [1,Theorem 3.32] and [2]). Theorem 1.6. Let G be a t.d.l.c. group and (g n ) be a net of elements of G converging to some limit g ∈ G. Then for n sufficiently large, there exist elements t n and r n of G such that t n , r n → 1 as n → ∞ and such that con(g n ) = t n con(g)t −1 n ; nub(g n ) = r n nub(g)r −1 n .
Denoting by S(G) the compact space of closed subgroups of G endowed with the Chabauty topology, we obtain the following consequence.
are both continuous.

Preliminaries
Relatively little is known about the overall topological dynamics of a t.d.l.c. group acting on itself. By contrast, there is a well-developed theory of the dynamics of Z acting on a t.d.l.c. group by continuous automorphisms. We recall some basic concepts and results in this area which will be needed in the sequel; the main references are [1], [7] and [10]. Notice that the results from [1] are stated under the hypothesis that the ambient groups are metrisable; this hypothesis has been removed by Jaworski [2]; we shall therefore freely refer to the results of [1] without any further comment about metrisability.
For the rest of this section, fix a t.d.l.c. group G. We only allow automorphisms that preserve the topology of G.
Given an automorphism f of G, the contraction group con(f ) of f on G is given by One sees that con(f ) is a subgroup of G, although not necessarily a closed subgroup.
Given a subgroup U of G, define the following subgroups: A tidy subgroup for f is an open compact subgroup of G that is tidy both above and below for f .
There is an open compact subgroup of G that is tidy for f .
(iii) We have U −− = con(f )U 0 (and similarly U ++ = con(f −1 )U 0 ). In particular, con(f ) = con(f −1 ) = 1 if and only if the set of open compact subgroups of G that are f -invariant forms a base of neighbourhoods of the identity.
Proof. For (i) and (ii), see [7] and [8]. The first assertion of part (iii) is [1, Proposition 3.16]. The second follows from the first, combined with (i). Part (iv) follows The proof for U + is similar.
A more recent development in the dynamical theory of automorphisms of t.d.l.c. groups is the nub of an automorphism. As usual, we will define the nub of a group element to be the nub of the corresponding inner automorphism. Before presenting several characterisations of the nub, we introduce one more subgroup associated with an automorphism f of a t.d.l.c. group G, namely the parabolic group par(f ). It is defined to consist of those elements x ∈ G such that the set {f n (x) | n ∈ N} is relatively compact. By [7, Proposition 3], the parabolic subgroup par(f ) is always closed.
Theorem 2.2. Let G be a t.d.l.c. group and let f be an automorphism of G. Then there is an f -invariant compact subgroup nub(f ) of G, which is equal to each of the following sets: (i) the intersection of all tidy subgroups for f ; (ii) the closure of the set bco(f ) := con(f ) ∩ par(f −1 ); (iii) the intersection con(f ) ∩ par(f −1 ); (iv) the intersection con(f ) ∩ con(f −1 ); For each g ∈ G, we denote by con(g/N) the inverse image in G of the contraction group of gN in G/N. By [1, Theorem 3.8], we have con(g/N) = con(g)N. Therefore, if con(g/N)/N is non-trivial, then con(g) is not contained in N. In particular, if G/N is not anisotropic, then G † is not contained in N.

Limits of contraction groups
The goal in this section is prove Theorem 1.6 from the introduction. This will be achieved after a series of technical preparations.
We start with a lemma which is a more precise version of [7, Lemma 14]. The lemma is used there in the proof that, if U is tidy for g then it is also tidy for every element in gU and that the scale is constant on gU.
Lemma 3.1. Let g ∈ G and let U be an open compact subgroup of G that is tidy above for g. Then for every u ∈ U there is t ∈ U + ∩ con(g −1 ) such that, for every Proof. It will be shown by induction that for every n ≥ 0 there is t n ∈ U + with t −1 n (gu) k t n = b n,k g k , with b n,k ∈ U, for every k ∈ {0, 1, . . . , n}.
The base case of the induction, when n = 0, is certainly true. Suppose that (1) has been established for n. The element t n+1 will be constructed as t n+1 = t n y, where y ∈ g −n U + g n .
Notice first that, for any choice of y ∈ g −n U + g n and all k ∈ {0, 1, . . . , n}, we have where b n+1,k = y −1 b n,k g k yg −k belongs to U. Now, to find suitable y, consider t −1 n+1 (gu) n+1 t n+1 = y −1 t −1 n (gu)(gu) n t n y = y −1 t −1 n g(ut n b n,n )g n y.
Since ut n b n,n belongs to U which is tidy above for g, we have ut n b n,n = w − w + with w ± ∈ U ± . Put y = g −n w −1 + g n . Then Since U + is compact, the sequence (t n ) n∈N has a subsequence converging to an element s ∈ U + . We see that for this choice of s. Moreover, note that we can freely replace s with sv given some v ∈ U 0 , since U 0 is normalised by g and hence [g l , v] ∈ U for all l ∈ Z. Now by Theorem 2.1(iv), we have hence by a suitable choice of v, we obtain t = sv ∈ U + ∩ con(g −1 ) with the desired property.
Proof. For k ≥ 0, there is b k ∈ U such that (gu) k t = tb k g k . Therefore, for each c ∈ con(g), we have The right side converges to the identity as k → ∞ because tb k belongs to U, which has a neighbourhood base of the identity comprising normal subgroups. Hence tcon(g)t −1 ≤ con(gu). Similarly, given c ∈ con(gu), then, for k ≥ 0, showing that t −1 ct ∈ con(g), so con(g) ≥ t −1 con(gu)t and hence tcon(g)t −1 = con(gu). Lemma 3.3. Let g ∈ G and let U be an open compact subgroup of G that is tidy above for g. Then for every u ∈ U ∩ g −1 Ug there is r ∈ U such that, for every k ∈ Z, Proof. Applying Lemma 3.1 yields t ∈ U + such that (2) holds for k ≥ 0 with r = t.
Applying the lemma a second time replacing g by g −1 and u by gu −1 g −1 , which belongs to U, yields s ∈ U − such that (2) holds for k ≤ 0 with r = s.
which belongs to U, and for k ≤ 0 If u n → 1 G , then r n with r n nub(g)r −1 n = nub(gu n ) may be chosen for each n such that r n → 1 G as n → ∞.
Proof of Theorem 1.6. Set u n := g −1 g n , so that u n → 1 as n → ∞. The tidying procedure (Theorem 2.1(i)) ensures that every open neighbourhood of the identity contains a compact open subgroup that is tidy above for g. In particular we may find a (possibly uncountable) chain (U n ) of open compact subgroups, all tidy above for g, with ∞ n U n = 1. Upon replacing (g n ) by a subnet, we may assume that u n ∈ U n for all n. Now Corollary 3.2 yields elements t n ∈ (U n ) + such that t n con(g)t −1 n = con(gu n ) = con(g n ). We have t n → 1 as n → ∞ since (U n ) + is contained in U n . Similarly, Corollary 3.4 yields elements r n ∈ G such that r n nub(g)r −1 n = nub(gu n ) = nub(g n ) and such that r n → 1 as n → ∞.

Normal closures
We now establish a connection between the abstract normal closure of an element f and the topological closure of its contraction group.
Proposition 4.1. Let G be a t.d.l.c. group and let A be an abstract subgroup of G. Let g ∈ A and suppose that K ≤ N G (A), where K is a closed subgroup of con(g) such that gKg −1 = K. Then K ≤ A. In particular, any (abstract) normal subgroup of G containing g also contains con(g).
Proof. The cyclic group generated by g is discrete or relatively compact. In the latter case, con(g) = 1. The desired assertion is thus empty in this case. We assume henceforth that the cyclic group generated by g is discrete. This implies that the subgroup H = K ∪ {g} is closed in G and splits as Let N be the (abstract) normal closure of g in H. From now on we may assume H = G, so that K = con(g).
Any element h ∈ H acting as a hyperbolic isometry fixes exactly two ends of T ; one of them must thus be ξ − . Since H is transitive on ∂T \ {ξ − }, it follows that some conjugate of h is contained in H ξ + . We have seen that N contains nub(g)⋊ g = H ξ + . We infer that N contains all elements of H acting as hyperbolic isometries on T .
Let now η ∈ ∂T \ {ξ − }. There is some h ∈ H such that ρ(h).ξ + = η. Using again the fact that ρ(H) fixes ξ − , we remark that if h is not a hyperbolic isometry, then hg is a hyperbolic isometry. Moreover we have ρ(hg).ξ + = η. Recalling that N contains all hyperbolic isometries of H, we infer that N is transitive on ∂T \ {ξ − }. Therefore N = H since N also contains H ξ + .
This proves that the normal closure of g in H contains K. Since by assumption H normalises A, we conclude that K ≤ A.
By combining Corollary 3.2 and Proposition 4.1, we arrive at a proof of Theorem 1.1.
Proof of Theorem 1.1. Let g ∈ G. Since D is dense in G, there is a net (g n ) in D that converges to g. By hypothesis, we have con(g n ) ≤ G † ≤ N G (D) for all n. Applying Proposition 4.1 with K = con(g n ) and A = D, we infer that con(g n ) is contained in D for all n. Moreover, by Corollary 3.2, for some n there is t ∈ con(g −1 ) such that tcon(g)t −1 = con(g n ), and consequently tcon(g)t −1 = con(g n ). Since t ∈ con(g −1 ) ≤ N G (D) by assumption, it follows that con(g) ≤ D. As g was arbitrary, we conclude that G † ≤ D.
We point out the following refinement of Corollary 1.4.
Corollary 4.2. Let G be a t.d.l.c. group. Then the following are equivalent: (i) G is topologically simple, and con(g) = 1 for some g ∈ G; (ii) G † is abstractly simple and dense in G, and the centre of G is trivial.
Proof. Suppose that G is topologically simple and that con(g) = 1 for some g ∈ G.
Evidently G is non-abelian, so Z(G) is a proper closed normal subgroup of G, hence Z(G) = 1. Moreover, G † is a non-trivial normal subgroup of G, and in fact G † must be dense in G by topological simplicity. Let N be a non-trivial normal subgroup of G † . Then N G (N ) ≥ (G † ) = G, since the normaliser of any closed subgroup of G is closed; thus N is a dense subnormal subgroup of G, and thus N = G † by Theorem 1.1. Hence G † is abstractly simple.
Conversely, suppose that G † is abstractly simple and dense in G, and that Z(G) = 1. Then in particular G † is non-trivial, so con(g) = 1 for some g ∈ G. Moreover, given a closed normal subgroup N of G, then either N ∩ G † = 1 or N ≥ G † , and in the latter case N = G. But if N ∩ G † = 1, then G † ≤ C G (N), so in fact N is central in G, using the fact that the centraliser of a subgroup is closed, and hence N is trivial by the assumption that Z(G) = 1. Hence G is topologically simple.