Related papers: Fundamental Groups and the Milnor Conjecture

Kontsevich conjectured that $\text{BDiff}(M, \text{rel }\partial)$ has the homotopy type of a finite CW complex for all compact $3$-manifolds with non-empty boundary. Hatcher-McCullough proved this conjecture when $M$ is irreducible. We…

We provide a geometric characterization of manifolds of dimension 3 with fundamental groups of which all conjugacy classes except 1 are infinite, namely of which the von Neumann algebras are factors of type $II_1$: they are essentially the…

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a…

In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal…

We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results…

Let $G$ be a connected compact Lie group, and let $M$ be a connected Hamiltonian $G$-manifold with equivariant moment map $\phi$. We prove that if there is a simply connected orbit $G\cdot x$, then $\pi_1(M)\cong\pi_1(M/G)$; if additionally…

We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete.…

In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite…

We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that `strongly converge' to the regular representation of the group. The corresponding statement…

It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply…

In the following work, an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented. This…

For $\rho, v>0$, we say that an $n$-manifold $M$ satisfies local $(\rho,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_\rho(\tilde x))\ge v>0$, where $\tilde x$ is an…

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the…

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the…

We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as "medium-scale" because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for…

Around twenty years ago Ghys conjectured that finite subgroups of the diffeomorphism group of a compact smooth manifold M have an abelian normal subgroup of index at most a(M), where a(M) depends only on M. First we construct a family of…

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with…

The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{g}) plays an important role in both geometric measure theory and the study of Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and with bounded…

A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable…

We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact…