If there exists an n-dimensional EG (that is, a contractible n-dimensional G-CW-complex on which G acts freely), we say that the geometric dimension of G  is at most n.  In a natural way, this defines the geometric dimension of G,
gd G \in \mathbb{N}\cup\{\infty\}.

It is easy to show that gd G \ge cd_R G, for each ring R.

Notice that Theorem 2.4 (in the case where each \cal X_\alpha consists of the trivial subgroup of G) says that, if G is the union of a well ordered continuous chain of subgroups of geometric dimension at most  n, then G itself has geometric dimension at most n+1.

It  follows, by induction on \aleph-rank(G), that, if

      \aleph-rank(H) \le r  =>  gd H \le  n, for each subgroup H of G,

then

      gd G  \le  \aleph-rank(G) -  r  +  n.

For r = -1, this says that, if G  is locally of geometric dimension at most  n, then

      gd G  \le  \aleph-rank(G) + 1 + n.

For n = 1, this says that, if G is locally free, then gd G \le \aleph-rank(G) + 2.   This may be an inefficient estimate as it is an open problem to decide whether every  locally free group has geometric dimension at most 2.

Similarly, if there is an n-dimensional EG (that is, a contractible n-dimensional G-CW-complex on which G acts with finite stabilizers, so that the fixed point set for every finite subgroup is contractible), say that the proper geometric dimension of G or gd G  is at most n.

It may be shown that this satisfies gd G  \ge cd_{\mathbb{Q}}G, where \mathbb{Q} denotes the rationals.

Theorem 2.4 (in the case where each \cal X_\alpha consists of all the finite subgroups of G_\alpha) says that, if G is the union of a well ordered continuous chain of subgroups of proper geometric dimension at most  n, then G itself has proper geometric dimension at most n+1.

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