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  \huge{The Department of Mathematics}\\[0.1\baselineskip]
  \Large{2021--22--A term}\\[0.2\baselineskip]

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\begin{description}
  \item[Course Name]
    Introduction to Geometric Group Theory

  \item[Course Number]
    \LRE{201‭.1‭.0311}

  \item[Course web page]\mbox{}\\
    \url{https://www.math.bgu.ac.il//en/teaching/fall2022/courses/introduction-to-geometric-group-theory}

  \item[Lecturer]
    Prof. Yair Glasner,
    \nolinkurl{<yairgl@bgu.ac.il>},
    Office 205

\item[Office Hours]
  \url{https://www.math.bgu.ac.il/en/teaching/hours}
\end{description}

\section*{Abstract}




\section*{Requirements and grading\footnote{Information may change during the first two weeks of the term. Please consult the webpage for updates}}






\section*{Course topics}

Basic concepts: Group actions, Cayley graphs and Schreier graphs, The word metric. Quasi isometries. The Milnor-Svarc lemma.
Free groups and trees: Group presentations. The groupAut(T),elliptic and hyperbolic elements. The boundary of the tree. Covering theory of graphs and the Nielsen Shcreier theorem. The ping-pong lemma. Free and amalgamated products, HNN extensions. The group PSL\_2 (Z)=Z/2Z*Z/3Z and its action on the Farey tree.
Some Hyperbolic geometry: Poincare models, their boundaries. Isometry groupsPSL\_2 (R),PSL\_2 (C), elliptic, hyperbolic, parabolic and loxodromic isometries and the trace. Free subgroups using the ping-pong lemma. Poincare lemma and surface groups as crystallographic hyperbolic groups. The fundamental domain of SL\_2 (Z)and the space of lattices. The Farey tessellation and continued fractions.
Hyperbolic groups.  Gromov hyperbolic spaces and their boundaries.  Hyperbolic groups, elliptic and hyperbolic elements. Quasi convex subgroups. Existence of free subgroups using ping-pong. Small cancellation groups. Solvability of the word problem and finite presentability. Existence of many quotients.

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