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We study complete, connected and simply connected $n$-dim Riemannian manifold $M$ satisfying Ricci curvature lower bound; moreover, suppose that it admits discrete isometric group actions $G$ so that the diameter of the quotient space $\mathrm{diam}(M/G)$ is bounded. In particular, for any $n$-manifold $M'$ satisfying $\mathrm{diam}(M') \le D$ and $\mathrm{Ric} \ge -(n-1)$, the universal cover and fundamental group $(\tilde{M}',G)$ satisfies the above condition. Let $\{(M_i,p_i)\}_{i \in \mathbb{N}}$ be a sequence of complete, connected and simply connected $n$-dim Riemmannian manifolds satisfying $\mathrm{Ric} \ge -(n-1)$. Let $G_i$ be a discrete subgroup of $\mathrm{Iso}(M_i)$ with $\mathrm{diam}(M_i/G_i) \le D$ where $D>0$ is fixed. Passing to a subsequence, $(M_i, p_i,G_i)$ equivariantly pointed-Gromov-Hausdorff converges to $(X,p,G)$. Then $G$ is a Lie group by Cheeger-Colding and Colding-Naber. We shall show that the identity component $G_0$ is a nilpotent Lie group. Therefore there is a maximal torus $T^k$ in $G$. Our first main result is that the fundamental group $\pi_1(X,p)$ is generated by loops in the $T^k$ orbit of $p$. In particular, $\pi_1(X)$ is a finitely generated abelian group. Assume that $M$ is a complete, connect and simply connected $n$-manifold with $\mathrm{Ric} \ge 0$ and $G$ is a discrete subgroup of $\mathrm{Iso}(M)$ with $\mathrm{diam}(M/G) \le D$. The celebrated splitting theorem by Cheeger-Gromoll shows that $M$ splits, $M=\mathbb{R}^k \times N$, where $N$ is a simply connected compact $(n-k)$-manifold. Our second result is that $\mathrm{diam}(N) \le D'(n,D)$.

It was shown by Mondino-Wei that any $\mathrm{RCD}^{*}(K,N)$ space $(X,d,\mathfrak{m})$ has a universal cover. We prove that for any point $x \in X$ and $R>0$, there exists $r

Let $(X,p)$ be a Ricci limit space. We show that for any $\epsilon > 0$ and $x \in X$, there exists $r< \epsilon$, depending on $\epsilon$ and $x$, so that any loop in $B_{r}(x)$ is contractible in $B_{\epsilon}(x)$. In particular, $X$ is semi-locally simply connected. Then we show that the generalized Margulis lemma holds for Ricci limit spaces of $n$-manifolds.

We study the topology of a Ricci limit space $(X,p)$, which is the Gromov-Hausdorff limit of a sequence of complete $n$-manifolds $(M_i, p_i)$ with $\mathrm{Ric}\ge -(n-1)$. Our first result shows that, if $M_i$ has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then $X$ is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of $X$ if $M_i$ has a uniform diameter bound.

Twisted light has recently gained enormous interest in communication systems. Thus far, twisted light has not yet been utilized for visible light communication to transmit data. Here, by exploiting the color and orbital angular momentum (OAM) degrees of freedom simultaneously, we construct a much higher-dimensional space spanned by their hybrid mode basis, which further increases the information capacity of twisted light. We build a new visible light communication system using a white light emitting diode, with red, green and blue (RGB) colors serving as independent channels and with OAM superposition states encoding the information. We connect our conceptually new RGB-OAM hybrid coding with the specially designed two-dimensional holographic gratings based on theta-modulation. After indoor free-space transmission, we decode the color information with an Xcube prism and subsequently decode the OAM superposition states with a pattern recognition method based on supervised machine learning. We succeed in demonstrating the transmission of color images and a piece of audio with the fidelity over 96%. Our point-to-point scheme with hybrid RGB-OAM encoding, not only increases significantly the information capacity of twisted light, but also offers additional security that supplements the traditional broadcasting visible light communications, e.g., Li-Fi.