Lower Semicontinuity of the Fundamental Group and Convergence with Discrete Symmetry.

References (in order of appearance in the video):

Cassorla, M. (1992). Approximating compact inner metric spaces by surfaces. Indiana University Mathematics Journal, 505-513.

Ferry, S. C., & Okun, B. L. (1995). Approximating topological metrics by Riemannian metrics. Proceedings of the American Mathematical Society, 123(6), 1865-1872.

Sormani, C., & Wei, G. (2001). Hausdorff convergence and universal covers. Transactions of the American Mathematical Society, 353(9), 3585-3602.

Sormani and Wei have published some papers about the theory of covering maps in length spaces:

Tuschmann, W. (1995). Hausdorff convergence and the fundamental group. Mathematische Zeitschrift, 218(1), 207-211.
The PhD Thesis referenced by Tuschmann (you will not find it online):
Rakotoniaina, C. (1985). Dégénérescence et sphères (Doctoral dissertation).

Example of polynomial growth groups:
Gromov, M. (1981). Groups of polynomial growth and expanding maps. Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 53(1), 53-78.
Pansu, P. (1983). Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergodic Theory and Dynamical Systems, 3(3), 415-445.

The results I mention about the nilpotent Lie groups throughout the talk can be found in
Clement, A. E., Majewicz, S., & Zyman, M. (2017). The theory of nilpotent groups (Vol. 43). Springer International Publishing.
Corwin, L., & Greenleaf, F. P. (2004). Representations of nilpotent Lie groups and their applications: Volume 1, Part 1, Basic theory and examples (Vol. 18). Cambridge university press.

On the virtual nilpotency of fundamental groups of spaces with Ricci curvature bounded below:
Kapovitch, V., & Wilking, B. (2011). Structure of fundamental groups of manifolds with Ricci curvature bounded below. arXiv preprint arXiv:1105.5955.
Breuillard, E., Green, B., & Tao, T. (2012). The structure of approximate groups. Publications mathématiques de l'IHÉS, 116(1), 115-221.
Much earlier, a result in this direction was found by Jeff Cheeger and Tobias Colding:
Cheeger, J., & Colding, T. H. (1995). Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below. Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 320(3), 353-357.

Turing, A. M. (1938). Finite approximations to lie groups. Annals of Mathematics, 105-111.

Jordan's Theorem and further topics can be found in:
Gelander, T. (2014). Lectures on lattices and locally symmetric spaces. arXiv preprint arXiv:1402.0962.

Gelander, T. (2012). A metric version of the Jordan--Turing theorem (No. arXiv: 1205.6553).

Montgomery, D., & Zippin, L. (2018). Topological transformation groups. Courier Dover Publications.

The elementary theory of ultrafilters and ultralimits can be found in:
Alexander, S., Kapovitch, V., & Petrunin, A. (2017). Alexandrov geometry. Book in preparation.
Breuillard, E., Green, B., & Tao, T. (2012). The structure of approximate groups. Publications mathématiques de l'IHÉS, 116(1), 115-221.

Berestovskii, V. N. (1989). Structure of homogeneous locally compact spaces with intrinsic metric. Siberian Mathematical Journal, 30(1), 16-25.

On the Gleason-Yamabe solution of Hilber's fifth problem, here are the original references. This topic has aged very well, and it is beautifully presented by Tao in his book.
Gleason, A. M. (1949). On the structure of locally compact groups. Proceedings of the National Academy of Sciences of the United States of America, 35(7), 384.
Yamabe, H. (1953). A generalization of a theorem of Gleason. Annals of Mathematics, 351-365.
Tao, T. (2014). Hilbert's fifth problem and related topics (Vol. 153). American Mathematical Soc.

Berestovskii, V. N. (1989). Homogeneous manifolds with intrinsic metric. II. Siberian Mathematical Journal, 30(2), 180-191.

Nagel, A., Stein, E. M., & Wainger, S. (1985). Balls and metrics defined by vector fields I: Basic properties. Acta Mathematica, 155(1), 103-147.

Le Donne, E., & Ottazzi, A. (2016). Isometries of Carnot groups and sub-Finsler homogeneous manifolds. The Journal of Geometric Analysis, 26(1), 330-345.

Gromov's Theorem on Almost Flat Manifolds. Peter Buser and Hermann Karcher wrote an entire book about this proof. In that book one can also find the proof of Malcev Theorem.
Gromov, M. (1978). Almost flat manifolds. Journal of Differential Geometry, 13(2), 231-241.
Buser, P., & Karcher, H. (1981). Gromov’s Almost Flat Manifolds. Soc. Math. de France. Astérisque Fascicules.

This video instersects largely with the manuscript:
Zamora, S. (2020). Fundamental Groups and Limits of Almost Homogeneous Spaces. arXiv preprint arXiv:2007.01985.