207,570 research outputs found

    Partial products in finite groups

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    Automorphisms of graph products of groups from a geometric perspective

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    This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product of certain elementary automorphisms (inner automorphisms, partial conjugations, automorphisms associated to symmetries of the underlying graph). This allows us to completely compute the automorphism group of certain graph products, for instance in the case where the underlying graph is finite, connected, leafless and of girth at least 55. If in addition the underlying graph does not contain separating stars, we can understand the geometry of the automorphism groups of such graph products of groups further: we show that such automorphism groups do not satisfy Kazhdan's property (T) and are acylindrically hyperbolic. Applications to automorphism groups of graph products of finite groups are also included. The approach in this article is geometric and relies on the action of graph products of groups on certain complexes with a particularly rich combinatorial geometry. The first such complex is a particular Cayley graph of the graph product that has a quasi-median geometry, a combinatorial geometry reminiscent of (but more general than) CAT(0) cube complexes. The second (strongly related) complex used is the Davis complex of the graph product, a CAT(0) cube complex that also has a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint arXiv:1803.07536. To appear in Proc. Lond. Math. So

    Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras

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    We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algebras, O(E,C)\mathcal{O}(E,C) and Or(E,C)\mathcal{O}^r(E,C), of a separated graph (E,C)(E,C), and of the abelianized Leavitt path algebra LKab(E,C)L_K^{\text{ab}}(E,C) as well. These algebras are the (reduced) crossed products with respect to the above-mentioned partial actions, and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary D∞D^{\infty}-saturated subsets of a certain infinite separated graph (F∞,D∞)(F_{\infty},D^{\infty}) built from (E,C)(E,C), called the separated Bratteli diagram of (E,C)(E,C). We finally use these tools to study simplicity and primeness of the tame separated graph algebras.Comment: 60 page

    Cartan subalgebras and the UCT problem, II

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    We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a suitable UHF algebra. These results rely on a new construction of Cartan subalgebras in certain inductive limits of Cartan pairs. We also provide a characterisation of the UCT problem in terms of finite order automorphisms, Cartan subalgebras and inverse semigroups of partial isometries of the Cuntz algebra O2\mathcal{O}_2. This generalizes earlier work of the authors.Comment: minor revisions; final version, accepted for publication in Math. Ann.; 26 page

    Spreading maps (polymorphisms), symmetries of Poisson processes and matching summation

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    The matrix of a permutation is a partial case of Markov transition matrices. In the same way, a measure preserving bijection of a space A with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) A to A, which spreads points of A into measures on A. In this paper, we discuss R-polymorphisms and ∨\vee-polymorphisms, who are analogues of the Markov transition operators for the groups of bijections A to A leaving the measure quasiinvariant; two types of the polymorphisms correspond to the cases, when A has finite and infinite measure respectively. We construct a functor from ∨\vee-polymorphisms to R-polymorphisms, it is described in terms of summation of convolution products of measures over matchings of Poisson configurations.Comment: 16 pages, European school on asymptotic combinatorics (St-Petersburg, July 2001
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