Category: Geometric Group Theory

G is a finitely presented group. X is its presentation complex (a simplicial 2-complex). Since G is finitely presented, the number of vertices of X is finite. Suppose \( u_1 , \cdots , u_q \) be the vertices of X. \( \tilde {X} \) be its universal cover. Fix lifts of the vertices of X.…

Here is the original paper: J. W. Alexander, On the deformation of an n-cell (A 2-page paper that influenced a remarkable amount of later work).

Motivation Start with a locally finite simplicial complex X. Locally finite: Each vertex is attached to only finitely many simplices. Why locally finite: To make sure it is a CW complex. Notice that the closure-finite criteria require each cell of a CW complex to meet only finitely many other cells. Hence we do not have a situation like…