Geodesic Spaces with Large Discrete Groups of Isometries
Open Access
- Author:
- Zamora Barrera, Sergio
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- April 01, 2022
- Committee Members:
- Nigel Higson, Major Field Member
Donald Richards, Outside Unit & Field Member
Anton Petrunin, Chair & Dissertation Advisor
Alexei Novikov, Major Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies - Keywords:
- Geometric Group Theory
Metric Geometry - Abstract:
- We use tools from geometric group theory to obtain results about metric spaces with large discrete groups of isometries. In the first part we obtain effective bounds for the diameters of universal covers. If $X$ is a compact semi-locally-simply-connected geodesic space with finite fundamental group $\pi_1(X)$, we show that the metric universal cover $\tilde{X}$ satisfies \[ \text{diam}(\tilde{X}) \leq 4 \sqrt{\vert \pi_1 (X) \vert } \cdot \text{diam}(X). \] If furthermore $X$ is a closed $n$-dimensional Riemannian manifold with finite abelian fundamental group $\pi_1(X)$, $Ric (X) \geq -(n-1)$, diameter $D$, and having a point with injectivity radius $\geq r_0 > 0$, then we show that its universal cover $\tilde{X}$ satisfies \[ \text{diam}( \tilde{X} ) \leq \left\lfloor 3 + \dfrac{3 \nu_n (2D+ r_0) }{2 \nu_n(r_0)} \log \vert \pi_1 (X) \vert \right\rfloor D, \] where $\nu_n(r) $ denotes the volume of a ball of radius $r$ in the $n$-dimensional hyperbolic space. In the second part, we show that in some sense the fundamental group is lower-semi-continuous for sequences of almost homogeneous spaces. That is, if we have a sequence $X_i$ of proper geodesic spaces and discrete groups of isometries $G_i \leq Iso (X_i)$ with diam$(X_i/G_i)\to 0$ converging in the pointed Gromov--Hausdorff sense to a semi-locally-simply-connected space $X$, then for large enough $i$, there are subgroups $\Lambda_i \leq \pi_1(X_i)$ and surjective morphisms \vspace{-0.15cm} \[ \Lambda_i \to \pi_1(X). \] In the third part, we show some restrictions on collapse under lower curvature bounds. We show that if a sequence $X_i$ of closed aspherical $n$-dimensional Riemannian manifolds of Ricci curvature bounded below and bounded diameter satisfies vol$(X_i) \to 0$, then for large enough $i$, there are non-trivial finitely generated abelian normal subgroups $1 \neq H_i \triangleleft \pi_1 (X_i)$. We also show that if a sequence $X_i$ of Riemannian manifolds homeomorphic to the $n$-dimensional torus and sectional curvature bounded below converges in the Gromov--Hausdorff sense to a $C^1$-Riemannian manifold X (possibly with non-empty boundary), then \[b_1(X )\geq \text{dim}(X),\] where $b_1(X)$ denotes the first Betti number of $X$.