Comeagre conjugacy classes and free products with amalgamation

More Polish full groups

We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an ω \omega -categorical structure can go down to reducts. Together, those results prove that a large number of ω \omega -categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any ω \omega -categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.

We show that all groups of a distinguished class of «large» topological groups, that of Roelcke precompact Polish groups, have Kazhdan’s Property (T). This answers a question of Tsankov and generalizes previous results by Bekka (for the infinite-dimensional unitary group) and by Evans and Tsankov (for oligomorphic groups). Further examples include the group \({{\,\mathrm{Aut}\,}}(\mu )\) of measure-preserving transformations of the unit interval and the group \({{\,\mathrm{Aut}\,}}^*(\mu )\) of non-singular transformations of the unit interval.More precisely, we prove that the smallest cocompact normal subgroup \(G^\circ \) of any given non-compact Roelcke precompact Polish group G has a free subgroup \(F\le G^\circ \) of rank two with the following property: every unitary representation of \(G^\circ \) without invariant unit vectors restricts to a multiple of the left-regular representation of F. The proof is model-theoretic and does not rely on results of classification of unitary representations. Its main ingredient is the construction, for any \(\aleph _0\)-categorical metric structure, of an action of a free group on a system of elementary substructures with suitable independence conditions.

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain nontrivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being nontrivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that [Formula: see text]-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking another approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a nontrivial system of linear identities, and obtain nontrivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every [Formula: see text]-categorical structure is homomorphically equivalent to a model-complete core.

The topic of this paper is the Mackey Cocycle Theorem: every Borel almost cocycle is equivalent to a Borel strict cocycle. This is a theorem about locally compact groups which is not true for arbitrary Polish groups. We discuss the theorem, the open question of whether the theorem generalizes to some nonlocally compact Polish groups, the generalization to non-Borel cocycles, and other subjects associated with the theorem. Traditionally, the subject of cocycles and related matters has been considered in the context of standard Borel G G -spaces. It is now known that a standard Borel G G -space has a topological realization as a Polish G G -space. This makes it possible to consider the subject from a topological point of view. The main theorem of this paper is that the conclusion of the Mackey Cocycle Theorem is equivalent to continuity properties of the almost cocycle. Even in the locally compact case, this continuity is a new result.

This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups.Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen-Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers.Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space C(G, ?) of all complex-valued functions on a finite group G. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside’s Theorem as an application of this theory—that any finite group of order paqb with p and q prime and with a+b > 1 has a nontrivial normal subgroup.Section 6 introduces cohomology of groups in connection with group extensions. If N is to be a normal subgroup of G and Q is to be isomorphic to G/N, the first question is to parametrize the possibilities for G up to isomorphism. A second question is to parametrize the possibilities for G if G is to be a semidirect product of N and Q.

The “peak reduction” method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was originally created by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well.In this paper, we present what seems to be the first applications of this technique in commutative algebra and affine algebraic geometry. We contribute toward a classification of two-variable polynomials by classifying, up to an automorphism, polynomials of the form \(a{\chi ^n} + b{y^m} + \sum\nolimits_{im + jn \leqslant mn} {cij{\chi ^i}} {y^j}\) (i.e., polynomials whose Newton polygon is either a triangle or a line segment). This has several applications to the study of embeddings of algebraic curves in the plane. In particular, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one can decide “almost” just by inspection whether or not a polynomial fiber {p(x, y) = 0} is an irreducible simply connected curve.Another application that we present here, yields a new decomposition of the group Aut(K(x, y)) in a free product with amalgamation.

A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every ${\aleph _0}$-stable, ${\aleph _0}$-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.

Bounded normal generation and invariant automatic continuity

Every Polish group is not free whereas some F σ group may be free. Also, every automorphism group of a structure of cardinality, e.g. ℶ ω , is not free.

We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.

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Introduction Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup theorem when a continuous Schreier transversal exists, and we give such a result in the category of Hausdorff /<Vgroups (Theorem 8). In the same category, we give an open subgroup version of the Kurosh Theorem (Theorem 13). The method of proof in both cases is a topological version of the groupoid method given by Higgins in [8]—that is, we use topological groupoids. The key steps are first of all to construct universal morphisms of topological groupoids, and secondly to prove that the pull-back by a covering morphism of a universal morphism is again universal. For the second step it is essential to know that the pullback of quotient maps of topological groupoids is again a quotient, and to obtain this we work in the category of Hausdorff /^-spaces. The chief technical work is then in constructing universal morphisms in this category—the results on /r^-spaces needed are Propositions A1-A3 which are given in the Appendix. This restriction to ^-spaces means that our theorems specialise only to countable versions of the abstract theorems. More general results can be given by using kgroupoids, but the proofs involve extra technicalities, and so we refer the interested reader to [7]. The authors wish to thank P. J. Higgins and P. Nickolas for helpful comments.

For a domain  in C, we denote by Aut() the group of holomorphic automorphisms of . It is obvious that Aut() is a topological group with respect to the law of composition and the compact-open topology. In particular, it is a theorem of H. Cartan that Aut() is in fact a Lie group if  is bounded. In light of the outstanding question “Which domains possess noncompact automorphism group?” there is much interest focused upon the existence and nonexistence of orbits of the automorphism group action accumulating at a given boundary point. The well-known Greene–Krantz conjecture belongs to such a line of research. In this paper, we discuss the finite-type boundary points that repel automorphism orbits. Denote by τ6(q) the D’Angelo type (see [10]) at q of the real hypersurface 6 in C. Now we present our main theorem.

It is proved that the tame automorphism group of a differential polynomial algebra k{x, y} over a field k of characteristic 0 in two variables x, y with m commuting derivations δ1, . . . , δm is a free product with amalgamation. An example of a wild automorphism of the algebra k{x, y} in the case of m ≥ 2 derivations is constructed.