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تسجيل مستخدم جديدLimits of contraction groups and the Tits core
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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.
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We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-
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Let $G$ be a Lie group over a totally disconnected local field and $alpha$ be an analytic endomorphism of $G$. The contraction group of $alpha$ ist the set of all $xin G$ such that $alpha^n(x)to e$ as $ntoinfty$. Call sequence $(x_{-n})_{ngeq 0}$ in
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A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends to infini
ty. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,f) which are central extensions of the additive group of the field of formal Laurent series over Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.
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In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of such a group
G, assuming that G is metrizable. (Recall that U_f consists of all group elements x such that f^n(x) tends to the identity element as n tends to infinity). Recently, Wojciech Jaworski showed that the main technical tool of the latter article remains valid in the non-metrizable case. He asserted without proof that, therefore, all results from that article remain valid. However, metrizability enters the arguments at a second point. In this note, we resolve this difficulty, by providing an affirmative answer to a question posed by Willis in 2004.
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