Narrow Results

In this paper, we present a resolution of discrete singular fibers of a closed 5-manifold equipped with a locally free S¹-action, and prove its compatibility with the resolution of cyclic surface singularities in the quotient orbifold by the S¹-action.

We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.

We classify actions of discrete abelian groups on some inclusions of von Neumann algebras, up to cocycle conjugacy. As an application, we classify actions of compact abelian groups on the inclusions of approximately finite dimensional (AFD) factors of type II₁ with index less than 4, up to stable conjugacy.

Let ${\mathrm{\text{AAut}}}_{\mathrm{\text{hol}}}(X)$ be the subgroup of the group ${\mathrm{\text{Aut}}}_{\mathrm{\text{hol}}}(X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under ${\mathrm{\text{AAut}}}_{\mathrm{\text{hol}}}(X)$ in terms of the dual graphs of the boundaries $\bar{X}\backslash X$ of their SNC-completions $\bar{X}$.

Let $A=\underrightarrow{lim}{A}_n$ be an AF algebra, $G$ be a compact group. We consider inductive limit actions of the form $\alpha =\underrightarrow{lim}{\alpha }_n$, where ${\alpha }_n:G\curvearrowright A_n$ is an action on the finite-dimensional C*-algebra $A_n$ which fixes each matrix summand. We give a complete classification up to conjugacy of such actions using twisted equivariant K-theory.

This paper gives a description of the diagonal ${\mathrm{\text{GL}}}_3$-orbits on the quadruple projective space ${({\mathbb{P}}^2)}^4$. We give explicit representatives of orbits, and describe the closure relations of orbits. A distinguished feature of our setting is that it is the simplest case, where diag$({\mathrm{\text{GL}}}_n)$ has infinitely many orbits but has also an open orbit in the multiple projective space ${({\mathbb{P}}^{n-1})}^m$.

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup , which has the property that partial actions of the group G give rise to actions of the inverse semigroup . We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev's original construction. We also show that almost invariant sets can be chosen to be in very good position.

We give a general structure theory for reconstructing non-trivial group actions on sets without any further assumptions on the group, the action, or the set on which the group acts. Using certain "local data" from the action we build a group of the data and a space with an action of on that arise naturally from the data . We use these to obtain an approximation to the original group G, the original space X and the original action G × X → X. The data is distinguished by the property that it may be chosen from the action locally. For a large enough set of local data , our definition of in terms of generators and relations allows us to obtain a presentation for the group G. We demonstrate this on several examples. When the local data is chosen from a graph of groups, the group is the fundamental group of the graph of groups and the space is the universal covering tree of groups. For general non-properly discontinuous group actions our local data allows us to imitate a fundamental domain, quotient space and universal covering for the quotient. We exhibit this on a non-properly discontinuous free action on ℝ. For a certain class of non-properly discontinuous group actions on the upper half-plane, we use our local data to build a space on which the group acts discretely and cocompactly. Our combinatorial approach to reconstructing abstract group actions on sets is a generalization of the Bass–Serre theory for reconstructing group actions on trees. Our results also provide a generalization of the notion of developable complexes of groups by Haefliger.

We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group $V$. The class of demonstrable groups for $V$ consists of all groups which can embed into $V$ with a natural dynamical behavior in their induced actions on the Cantor space ${ℭ}_2:={\{0,1\}}^{\omega }$. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while Thompson’s group $V$ is a candidate as a universal $co{𝒞}{ℱ}$ group by Lehnert’s conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of $co{𝒞}{ℱ}$ groups as it shows that all four of the known closure properties of the class of $co{𝒞}{ℱ}$ groups hold for the set of finitely generated subgroups of $V$.

We consider groups $G$ which have generating sets consisting of pairwise conjugate elements. We introduce a natural condition on such sets, namely path-connectedness, which has strong consequences when $G$ acts on a real tree. The main result of the paper states that the group of type-preserving automorphisms of a regular simplicial tree has a generating set with this property.

We use the set symmetric difference between vertex subsets of a finite undirected simple graph $G$ to define a binary operation ∘ on the vertex set of a new graph $G^{\S }$, that contains $G$ as subgraph and whose vertices are non-empty vertex subsets of $G$. We show how the binary operation ∘ determines an algebraic structure on $V(G^{\S })$ that is strictly related to the graph structure of $G^{\S }$. In fact, we show that $\mathrm{\text{Aut}}(V(G^{\S }),\circ )$ agrees with $\mathrm{\text{Aut}}(G^{\S })$ and, next, we provide several characterizations for the algebraic structure $(V(G^{\S }),\circ )$ when the graph $G^{\S }$ is connected and locally dissymmetric.

Let M, M′ be compact oriented 3-manifolds and L′ a link in M′. We prove that, under certain conditions, the topological types of M and (M′, L′) determine the degree of a cyclic covering p : M → M′, branched over L′.

For genus one knots the Casson-Gordon invariant τ is expressed in terms of the classical signatures, generalizing an earlier result of P. Gilmer. As an application it is shown that the pretzel knots K(3, –5, 7) and K(3, –5, 17) are not concordant to their reverses.

Let $G$ be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in $m$ commuting derivations equipped with a $G$-action by differential field automorphisms. In the language of $G$-differential rings (i.e. the language of rings with added symbols for derivations and automorphisms), we prove that this class has a model-companion — denoted $G-{\mathrm{\text{DCF}}}_{0,m}$. We then deploy the model-theoretic tools developed in the first author’s paper [D. M. Hoffmann, Model theoretic dynamics in a Galois fashion, Ann. Pure Appl. Logic 170(7) (2019) 755–804] to show that any model of $G-{\mathrm{\text{DCF}}}_{0,m}$ is supersimple (but unstable when $G$ is nontrivial), a PAC-differential field (and hence differentially large in the sense of the second author and Tressl [Differentially large fields, preprint (2020), arXiv:2005.00888, available at https://arxiv.org/abs/2005.00888]), and admits elimination of imaginaries after adding a tuple of parameters. We also address model-completeness and supersimplicity of theories of bounded PAC-differential fields (extending the results of Chatzidakis and Pillay [Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998) 71–92] on bounded PAC-fields).

In this paper, we give a detailed proof of a result due to Torsten Ekedahl, describing complex tori admitting a rigid group action and showing explicitly their projectivity and their structure in terms of CM-fields. In the appendix, joint with Claudon, we show, using a method of Green-Voisin, that all group actions on complex tori deform to projective ones.

Let $\rho :\mathbb{Z}/k\mathbb{Z}\to SL(2,ℂ)$ be a representation of a finite abelian group and let ${\mathrm{\Theta }}^{\text{gen}}\subset {\mathrm{\text{Hom}}}_{\mathbb{Z}}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of generic stability conditions on the set of $G$-constellations. We provide a combinatorial description of all the chambers $C\subset {\mathrm{\Theta }}^{\mathrm{\text{gen}}}$ and prove that there are $k!$ of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric $G$-constellations, it is enough to build all simple chambers. We also prove that there are $k\cdot 2^{k-2}$ simple chambers. Finally, we provide an explicit formula for the tautological bundles ${{ℛ}}_C$ over the moduli spaces ${{ℳ}}_C$ for all chambers $C\subset {\mathrm{\Theta }}^{\mathrm{\text{gen}}}$ which only depends upon the chamber stair which is a combinatorial object attached to the chamber $C$.

In this paper, the mean values of the recurrence are computed for general group actions. Let $(X,d)$ be a metric space with a finite measure $\mu$ and $G$ be a countable group acting on $(X,d)$. Let $F_1,F_2,\dots$ be a sequence of subsets of $G$ with $\left|F_n\right|\to \infty$ and put $E_n=F_n^{-1}{F}_n$. If the Hausdorff measure $H_h$ is finite on $X$ and $\mu$ is $T$-invariant. We assume that $\mu$ and $H_h$ are concordant. Then the function

We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions.

We develop the concept of a double (more generally $n$-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.