# at.algebraic topology – Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $$L$$ of a nested sequence of compact 2-dimensional polyhedra
$$L_0 to L_1 to L_2 to cdots$$
obtained as follows. Take $$L_0$$ as some $$S_g$$, an oriented compact surface of positive
genus $$g$$ from which an open disk has been deleted. To form $$L_{n+1}$$ from $$L_n$$, for each
loop $$a$$ in $$L_n$$ that generates the first homology group $$H_1(L_n)$$, attach to $$L_n$$ some $$S_{g_a}$$ by identifying
the boundary of $$S_{g_a}$$ with the loop $$a$$. Since the fundamental group of $$S_g$$ punctured
is a free group on $$2g$$ generators, this procedure embeds each $$pi_1(L_n)$$, and thus
each finitely generated subgroup of $$pi_1(L)$$, as a subgroup of a free group, with each
generator $$a$$ of $$pi_1(L_n)$$ becoming a product of $$g_a$$ commutators in $$pi_1(L_{n+1})$$. Hence
$$pi_1(L)$$ is a countable, perfect, locally free group. See What is a grope? for some motivation.

Here we build a grope $$A$$ in stages. At each stage put in disks with 3 handles on all the various handle curves at that stage. In other words, grope $$A$$ is built using disks with 3 handles uniformly throughout.

Question. Is it possible that there is a generating set $$G$$ of $$pi_1(A)$$ such
that each element of $$G$$ has commutator length bounded by 2?

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