The "Asymptotic dimension and some applications" seminar was held once a week, for 80 minutes, in Ben-Gurion University Mathematics Department.
There were 13 seminars held in total during the Spring 2013 semester.
The level of exposition of the content was approachable for graduate students who have finished a topology course.
You can find my handwritten notes for all of the seminars attached below. Please let me know if you see any mistakes.
Here is the plan for the seminar, including a list of sources. ( A pdf version of this list is attached below).
For the beginning:
to see the equivalent defnitions, some examples and some properties of asymptotic dimension, we used the following two survey papers:
G. Bell, A. N. Dranishnikov, Asymptotic Dimension in Bedlewo, Topology Proc. 38 (2011), 209-236, arXiv:math/0507570 (arXiv version from 2005)
G. Bell, A. N. Dranishnikov, Asymptotic Dimension, Topology and its Appls. 155 (2008), 1265-1296, arXiv:math/0703766 (arXiv version from 2007)
Additional sources for properties of asymptotic dimension, and asdim of groups:
A. N. Dranishnikov, J. Smith, Asymptotic dimension of discrete groups, arXiv:math/0603055
J. Smith, On asymptotic dimension of countable abelian groups, arXiv:math/0504447
N. Brodskiy, J. Dydak, M. Levin, A. Mitra, Hurewicz theorem for Assouad-Nagata Dimension, arXiv:math/0605416v2
Main objective:
to see the proof of the following formula: the asymptotic dimension of every (finitely generated) hyperbolic group G equals the covering dimension of its boundary plus 1, that is, asdim G = dim(\partial G) + 1.
(In fact, this formula works for all spaces that are hyperbolic, geodesic, proper and cobounded, but hyperbolic groups are a neat example of these.)
Main source for this: S. Buyalo, V. Schroeder, Elements of asymptotic geometry, European Mathematical Society, 2007
The file with corrected proof of Theorem 12.2.1 from Buyalo-Schroeder's book can be found below, titled Thm12.2.1-corrected-proof-Lebedeva-2019-edited.
Alternative sources for this (most are available on arXiv):
S. Buyalo and N. Lebedeva, Dimension of locally and asymptotically self-similar spaces, St. Petersburg Math. Jour. 19 (2008), 45-65, russian version: Algebra i analiz 19 (1) (2007), 60-92, arXiv:math/0509433 (arXiv version from 2005)
S. Buyalo, Capacity dimension and embedding of hyperbolic spaces into the product of trees, St. Petersburg Math. Jour. 17 (2006), 581-591, russian version: Algebra i analiz 17 (4) (2005), 39-55, arXiv:math/0505429
S. Buyalo and V. Schroeder, Embedding of hyperbolic spaces in the product of trees, Geom. Dedicata, 113 (2005), 75-93, arXiv:math/0311524 (arXiv version from 2003)
S. Buyalo, A. Dranishnikov and V. Schroeder, Embedding of hyperbolic groups into products of binary trees, Invent. Math. 169 (1) (2007), 153-192 .
S. Buyalo, Asymptotic dimension of a hyperbolic space and capacity dimension of its boundary at infinity, St. Petersburg Math. Jour. 17 (2006), 267-283, russian version: Algebra i analiz 17 (2) (2005), 70-95, arXiv:math/0505427
Other sources we used, for facts about hyperbolic groups, hyperbolic spaces, boundary at infinity and coarse geometry:
J. Alonso, T. Brady et al., Notes on word hyperbolic groups
M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funkt. Anal. 10 (2000), no. 2, 266--306.
M. Bridson and A. Haefliger, Metric spaces of nonpositive curvature, Springer Verlag, Berlin, 1999
C. Drutu and M. Kapovich, Lectures on the geometric group theory
E. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d'apres Mikhael Gromov, Progr. Math. vol. 83, Birkhauser, Boston, MA, 1990
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, vol. 2
I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, Contemp. Math. 296 (2002), 39-93.
U. Lang and T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 2005, no. 58, 3625-3655, arXiv:math/0410048
J. Roe, Lectures on coarse geometry, book by University Lecture Series, vol. 31, AMS , 2003