Searcharxiv⌕ Search

arXiv subjects

Shin-ichi Oguni

Publications and source records attributed to Shin-ichi Oguni.

17 recordsLinked to original sources
arXiv2025

Acylindrical hyperbolicity and the centers of Artin groups that are not free of infinity

Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them. The authors developed their study and clarified when acylindrical hyperbolicity holds for Artin groups of infinite type associated with graphs that are not cones. In this paper, we introduce reduced clique-cube complexes. By using them, we show acylindrical hyperbolicity of irreducible Artin groups associated with graphs that are not cliques, that is, irreducible Artin groups that are not free of infinity. Such Artin groups contain infinite type Artin groups of type FC. As an application, we see that the centers of such Artin groups are finite, and that actually they are trivial in many cases.

math.GR↗
arXiv2023

Acylindrical hyperbolicity of Artin groups associated with graphs that are not cones

Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and actions on them. In this paper, by developing their study and formulating some additional discussion, we demonstrate that acylindrical hyperbolicity holds for more general Artin groups. Indeed, we are able to treat Artin groups of infinite type associated with graphs that are not cones.

math.GT↗
arXiv2020

Acylindrical hyperbolicity of Artin-Tits groups associated to triangle-free graphs and cones over square-free bipartite graphs

It is conjectured that the central quotient of every irreducible Artin group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin groups associated to triangle-free graphs and Artin groups of large type associated to cones over square-free bipartite graphs. In fact, we treat Artin groups that are known to be CAT(0) groups by a result of Brady and McCammond.

math.GR↗
arXiv2018

Coarse compactifications and controlled products

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space complements the space as a coarse compactification. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

math.MG↗
arXiv2018

A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture

We establish a coarse version of the Cartan-Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum-Connes conjecture. Combined with the result of Osajda-Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum-Connes conjecture.

math.MG↗
arXiv2017

On coarse geometric aspects of the Hilbert geometry

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum-Connes conjecture holds for any two-dimensional Hilbert geometry via Yu's theorem.

math.MG↗
arXiv2014

The coarse Baum-Connes conjecture for Busemann non-positively curved spaces

We prove that the coarse assembly maps for proper metric spaces which are non-positively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces are with bounded coarse geometry. Also it is shown that we can calculate the coarse K-homology and the K-theory of the Roe algebra by using the visual boundaries.

math.KT↗
arXiv2014

Coronae of relatively hyperbolic groups and coarse cohomologies

We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the $K$-homology of the corona with the $K$-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the $K$-theory of the corona with the $K$-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the $K$-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

math.KT↗
arXiv2014

Coronae of product spaces and the Coarse Baum-Connes conjecture

We study the coarse Baum-Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum-Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, "corona", of a product of proper metric spaces.

math.KT↗
arXiv2013

Blowing up and down compacta with geometrically finite convergence actions of a group

We consider two compacta with minimal non-elementary convergence actions of a countable group. When there exists an equivariant continuous map from one to the other, we call the first a blow-up of the second and the second a blow-down of the first. When both actions are geometrically finite, it is shown that one is a blow-up of the other if and only if each parabolic subgroup with respect to the first is parabolic with respect to the second. As an application, for each compactum with a geometrically finite convergence action, we construct its blow-downs with convergence actions which are not geometrically finite.

math.GR↗
arXiv2013

On relative hyperbolicity for a group and relative quasiconvexity for a subgroup

We consider two families of subgroups of a group. Each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is supposed to be hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.

math.GR↗
arXiv2012

On Cannon-Thurston maps for relatively hyperbolic groups

Baker and Riley proved that a free group of rank 3 can be contained in a hyperbolic group as a subgroup for which the Cannon-Thurston map is not well-defined. By using their result, we show that the phenomenon occurs for not only a free group of rank 3 but also every non-elementary hyperbolic group. In fact it is shown that a similar phenomenon occurs for every non-elementary relatively hyperbolic group.

math.GR↗
arXiv2011

The coarse Baum-Connes conjecture for relatively hyperbolic groups

We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. Especially, the group satisfies the analytic Novikov conjecture.

math.KT↗