# Geometric group theory, an introduction [lecture notes] by Clara Löh

By Clara Löh

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Gn ∈ (G1 G2 )∗ with n ∈ N and g1 , . . , gn ∈ G1 G2 reduced, if for all j ∈ {1, . . , n − 1} • either gj ∈ G1 \ {e} and gj+1 ∈ G2 \ {e}, • or gj ∈ G2 \ {e} and gj+1 ∈ G1 \ {e}. Similarly, one can also describe free amalgamated products and HNNextensions by suitable classes of reduced words [119, Chapter I]. 1. Suppose the group F is freely generated by S.

4 (Isomorphism rigidity of Cayley graphs). , for which finitely generated groups G and H there exist finite generating sets S ⊂ G and T ⊂ H such that the graphs Cay(G, S) and Cay(H, T ) are isomorphic. , given (up to translation by a fixed group element) by a group isomorphism. Both of these questions ask for rigidity properties of Cayley graphs, namely, how much of the algebraic structure is rigid enough to be visible in the combinatorics of all Cayley graphs of a given group. These questions are well studied for finite groups.

Because n > 2, the two paths v0 , vn−1 and v0 , . . , vn−1 are different, and both connect v0 with vn−1 , which is a contradiction. Hence, X is a tree. Conversely, let X be a tree; in particular, X is connected, and every two vertices can be connected by a path in X. Assume for a contradiction that there exist two vertices v and v that can be connected by two different paths p and p . 2), contradicting the fact that X is a tree. Hence, every two vertices of X can by connected by exactly one path in X.

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