207,570 research outputs found
Automorphisms of graph products of groups from a geometric perspective
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 . 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
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,
and , of a separated graph , and
of the abelianized Leavitt path algebra 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 -saturated
subsets of a certain infinite separated graph built
from , called the separated Bratteli diagram of . 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
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 .
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
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 -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 -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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