12,831 research outputs found
Turbulence, amalgamation and generic automorphisms of homogeneous structures
We study topological properties of conjugacy classes in Polish groups, with
emphasis on automorphism groups of homogeneous countable structures. We first
consider the existence of dense conjugacy classes (the topological Rokhlin
property). We then characterize when an automorphism group admits a comeager
conjugacy class (answering a question of Truss) and apply this to show that the
homeomorphism group of the Cantor space has a comeager conjugacy class
(answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups
that admit comeager conjugacy classes in any dimension (in which case the
groups are said to admit ample generics). We show that Polish groups with ample
generics have the small index property (generalizing results of
Hodges-Hodkinson-Lascar-Shelah) and arbitrary homomorphisms from such groups
into separable groups are automatically continuous. Moreover, in the case of
oligomorphic permutation groups, they have uncountable cofinality and the
Bergman property. These results in particular apply to automorphism groups of
many -stable, -categorical structures and of the random
graph. In this connection, we also show that the infinite symmetric group
has a unique non-trivial separable group topology. For several
interesting groups we also establish Serre's properties (FH) and (FA)
Simultaneous dense and nondense orbits for commuting maps
We show that, for two commuting automorphisms of the torus and for two
elements of the Cartan action on compact higher rank homogeneous spaces, many
points have drastically different orbit structures for the two maps.
Specifically, using measure rigidity, we show that the set of points that have
dense orbit under one map and nondense orbit under the second has full
Hausdorff dimension.Comment: 17 pages. Very minor changes to the exposition. Three additional
papers cite
Braid groups of imprimitive complex reflection groups
We obtain new presentations for the imprimitive complex reflection groups of
type and their braid groups for . Diagrams
for these presentations are proposed. The presentations have much in common
with Coxeter presentations of real reflection groups. They are positive and
homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms
correspond to group automorphisms. The new presentation shows how the braid
group is a semidirect product of the braid group of affine type
and an infinite cyclic group. Elements of are
visualized as geometric braids on strings whose first string is pure and
whose winding number is a multiple of . We classify periodic elements, and
show that the roots are unique up to conjugacy and that the braid group
is strongly translation discrete.Comment: published versio
Automorphism Groups of Homogeneous Structures
A homogeneous structure is a countable (finite or countably infinite) first order structure such that every isomorphism between finitely generated substructures extends to an automorphism of the whole structure. Examples of homogeneous structures include any countable set, the pentagon graph, the random graph, and the linear ordering of the rationals. Countably infinite homogeneous structures are precisely the Fraisse limits of amalgamation classes of finitely generated structures. Homogeneous structures and their automorphism groups constitute the main theme of the thesis.
The automorphism group of a countably infinite structure becomes a Polish group when endowed with the pointwise convergence topology. Thus, using Baire Category one can formulate the following notions. A Polish group has generic automorphisms if it contains a comeagre conjugacy class. A Polish group has ample generics if it has a comeagre diagonal conjugacy class in every dimension. To study automorphism groups of homogeneous structures as topological groups, we examine combinatorial properties of the corresponding amalgamation classes such as the extension property for partial automorphisms (EPPA), the amalgamation property with automorphisms (APA), and the weak amalgamation property. We also explain how these combinatorial properties yield the aforementioned topological properties in the context of homogeneous structures.
The main results of this thesis are the following. In Chapter 3 we show that any free amalgamation class over a finite relational language has Gaifman clique faithful coherent EPPA. Consequently, the automorphism group of the corresponding free homogeneous structure contains a dense locally finite subgroup, and admits ample generics and the small index property. In Chapter 4 we show that the universal bowtie-free countably infinite graph admits generic automorphisms. In Chapter 5 we prove that Philip Hall's universal locally finite group admits ample generics. In Chapter 6 we show that the universal homogeneous ordered graph does not have locally generic automorphisms. Moreover we prove that the universal homogeneous tournament has ample generics if and only if the class of finite tournaments has EPPA
Reconstructing Structures with the Strong Small Index Property up to Bi-Definability
Let be the class of countable structures with the strong
small index property and locally finite algebraicity, and the
class of such that for every . For homogeneous , we introduce what we call the
expanded group of automorphisms of , and show that it is second-order
definable in . We use this to prove that for ,
and are isomorphic as abstract groups if and only if
and are isomorphic as permutation groups. In
particular, we deduce that for -categorical structures the
combination of strong small index property and no algebraicity implies
reconstruction up to bi-definability, in analogy with Rubin's well-known
-interpretation technique of [7]. Finally, we show that every
finite group can be realized as the outer automorphism group of for
some countable -categorical homogeneous structure with the strong
small index property and no algebraicity
Group elastic symmetries common to continuum and discrete defective crystals
The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of R 3 , with addition as group operation. We consider whether or not the symmetries of these discrete subgroups extend to symmetries of (particular) ambient Lie groups. It turns out that those symmetries which correspond to automorphisms of the discrete structures do extend to (continuous) symmetries of the ambient Lie group (just as the symmetries of a perfect lattice may be embedded in ‘homogeneous elastic’ deformations). Other types of symmetry must be regarded as ‘inelastic’. We show, following Kamber and Tondeur, that the corresponding continuous automorphisms preserve the Cartan torsion, and we characterize the discrete automorphisms by a commutativity condition, (6.14), that relates (via the matrix exponential) to the dislocation density tensor. This shows that periodicity properties of corresponding energy densities are determined by the dislocation density
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