Medium-scale curvature at larger radii in finitely generated groups

Cheeger-Gromoll Splitting Theorem for groups

We study a notion of curvature for finitely generated groups which serves as a role of Ricci curvature for Riemannian manifolds. We prove an analog of Cheeger-Gromoll splitting theorem. As a consequence, we give a geometric characterization of virtually abelian groups. We also explore the relation between this notion of curvature and the growth of groups.

Properties of linear groups with restricted unipotent elements

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero linear groups. In particular if such a linear group is finitely generated then centralisers virtually split and all finitely generated abelian subgroups are undistorted. If further the group is virtually torsion free (which always holds in characteristic zero) then we have a stron…

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ …

Negative curvature in automorphism groups of one-ended hyperbolic groups

In this article, we show that some negative curvature may survive when taking the automorphism group of a finitely generated group. More precisely, we prove that the automorphism group $\mathrm{Aut}(G)$ of a one-ended hyperbolic group $G$ turns out to be acylindrically hyperbolic. As a consequence, given a group $H$ and a morphism $φ: H \to \mathrm{Aut}(G)$, we deduce that the semidirect product $G \rtimes_φH$ is acylindrically hyperbolic if and only if $\mathrm{ker}(H \oversetφ{\to} \mathrm{Aut}(G) \to \mathrm{Out}(G))$ is finite.

On flat submaps of maps of non-positive curvature

We prove that for every $r>0$ if a non-positively curved $(p,q)$-map $M$ contains no flat submaps of radius $r$, then the area of $M$ does not exceed $Crn$ for some constant $C$. This strengthens a theorem of Ivanov and Schupp. We show that an infinite $(p,q)$-map which tessellates the plane is quasi-isometric to the Euclidean plane if and only if the map contains only finitely many non-flat vertices and faces. We also generalize Ivanov and Schupp's result to a much larger class of maps, namely to maps with angle functions.

Counting open negatively curved manifolds up to tangential homotopy equivalence

Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.

Cayley图的Bakry-Émery和Ollivier Ricci曲率

Bakry-Émery and Ollivier Ricci Curvature of Cayley Graphs

本文研究了Cayley图上的Bakry-Émery曲率和Ollivier Ricci曲率，引入RAACHs推广RAAGs和RACGs并推导特定曲率，同时证明一般有限展示群添加关系元不会降低其Cayley图的自适应加权曲率。