Search

Given a non-necessarily commutative unital ring R and a unital partial representation Θ of a group G into the Picard semigroup PicS (R) of the isomorphism classes of partially invertible R -bimodules, we construct an abelian group C (Θ / R) formed by the isomorphism classes of partial generalized crossed products related to Θ and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C 0 (Θ / R) of C (Θ / R). Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings R ⊆ S with the same unity and a unital partial representation G → S R (S) of an arbitrary group G into the monoid S R (S) of the R -subbimodules of S. This generalizes the works by Kanzaki and Miyashita. [ABSTRACT FROM AUTHOR]

We investigate sections of the arithmetic fundamental group $\pi _1(X)$ where X is either a smooth affinoid p-adic curve , or a formal germ of a p-adic curve , and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y , and the exact sequence of $\pi _1(X)$ splits , then $\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $\pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite. [ABSTRACT FROM AUTHOR]

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups A indexed by elements of ω ω . They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits lim n A must vanish, for n > 0. They also proved that under the Continuum Hypothesis lim 1 A ≠ 0. The question whether lim n A vanishes, for all n > 0 , has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under the Proper Forcing Axiom (PFA) lim 1 A = 0. Bergfalk showed that it is consistent that lim 2 A does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that lim n A = 0 , for all n. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Hanson. We complete the picture by showing that, for any n > 0 , it is relatively consistent with ZFC that lim n A ≠ 0. [ABSTRACT FROM AUTHOR]

This paper analyzes some of the fundamental results of the unit groups of the commutative modular group algebras. We introduce a more visible and concise form of a number of these results. We consider with a corresponding justification some essentially inexact and unproved results in some papers and for certain of them either we mark the corrections or we point out the corrections, which are founded in another papers. [ABSTRACT FROM AUTHOR]

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in \mathbb {Z}^{(\mathbb {N}\times [c])}, where c\in \mathbb {N}, has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature. [ABSTRACT FROM AUTHOR]

We study \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study the special value u=1 of Artin-Ihara L-functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study an important well-known structure operation, namely, the semidirect product (or split extensions) which is a very useful tool to structure certain kinds of groups. More precisely, we study the isomorphism problem for semidirect products and then we determine how isomorphism of semidirect products and conjugacy of the images of the corresponding actions are related. As an application, for two positive integers n and m , we compute the number of upper isomorphism classes of split extensions of an elementary abelian p -group of order p n by an elementary abelian p -group of order p m . Furthermore, we deal with split extensions where the kernel is a non-abelian p -group of nilpotency class two and the quotient is an elementary abelian p -group. [ABSTRACT FROM AUTHOR]

We are motivated by [M. Arkowitz. K. Maruyama. J. Math. Soc. Japan, 66(3):735-743, 2014]: “It would be interesting to compute other Gottlieb groups of Moore spaces such as, for example, Gn+1(M(A, n))” to compute the Gottlieb groups Gn+k(M(Z m +Z2, n)) for k = 1,2 and m ≥1. [ABSTRACT FROM AUTHOR]

We give a condition on the defining graph of a right-angled Artin group, which implies that its automorphism group is virtually indicable, that is, it has a finite index subgroup that admits a homomorphism onto Z. We use this as a part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence, we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group. [ABSTRACT FROM AUTHOR]

Associated to a convex integral polygon N is a cluster integrable system XN constructed from the dimer model. We compute the group GN of symmetries of XN , called the (2-2) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, non-torsion elements of GN are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, GN is a subgroup of the Picard group of a certain algebraic surface associated to N. [ABSTRACT FROM AUTHOR]