- #1
Bacle
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Hi, Everyone:
I am looking for a proof or ref. that every finitely-generated group G is the
fundamental group of a 2-cell complex.
This is either a corollary of Reidemeister-Schreier's theorem/method for producing
a presentation of a subgroup H from the presentation of the supergroup G , or, in
some versions, it is the actual theorem.
I am not sure of how it works, but I know the following, from what I read (a paper
with missing source, unfortunately):
I know we define a right action of G on a set V, where V are the vertices --0-cells--
by adjoining vertex vi with vertex vi.g ( so that we have a regular graph whose degree
is the cardinality of G ; every vertex v has |G| outgoing vertices {v.g:g in G} , and
|G| incoming vertices ; we join vg^-1 with v through g, since vg^-1g=v=v_id); then there is a vertex between v, vg_i, vg_ig_j,...; and between vg_k and {vg_k.g_j, vg_k.g_j.g_r ,..} , etc. To each path in the graph, we associate a word: the path joining , say, vgi to vgi.gj to vgi.gj.gk is assigned the word gi.gj.gk , etc.
Then the 1-complex, i.e., the edges describe the action, and the 2-cells are
used to describe the relations of G. The relations are elements of the stabilizer of
all x, i.e., a relation is a word in G that produces a loop at each vertex v (i.e.,
for every vg_k the relation-word {g_i1.g_i2...g_ik} sends every element into a loop,
so that, for all g_o in G , we get v_go.( g_i1.g_i2...g_ik)=v_go )
We then somehow use the relations as polygons.
Thanks for any Reference.
I am looking for a proof or ref. that every finitely-generated group G is the
fundamental group of a 2-cell complex.
This is either a corollary of Reidemeister-Schreier's theorem/method for producing
a presentation of a subgroup H from the presentation of the supergroup G , or, in
some versions, it is the actual theorem.
I am not sure of how it works, but I know the following, from what I read (a paper
with missing source, unfortunately):
I know we define a right action of G on a set V, where V are the vertices --0-cells--
by adjoining vertex vi with vertex vi.g ( so that we have a regular graph whose degree
is the cardinality of G ; every vertex v has |G| outgoing vertices {v.g:g in G} , and
|G| incoming vertices ; we join vg^-1 with v through g, since vg^-1g=v=v_id); then there is a vertex between v, vg_i, vg_ig_j,...; and between vg_k and {vg_k.g_j, vg_k.g_j.g_r ,..} , etc. To each path in the graph, we associate a word: the path joining , say, vgi to vgi.gj to vgi.gj.gk is assigned the word gi.gj.gk , etc.
Then the 1-complex, i.e., the edges describe the action, and the 2-cells are
used to describe the relations of G. The relations are elements of the stabilizer of
all x, i.e., a relation is a word in G that produces a loop at each vertex v (i.e.,
for every vg_k the relation-word {g_i1.g_i2...g_ik} sends every element into a loop,
so that, for all g_o in G , we get v_go.( g_i1.g_i2...g_ik)=v_go )
We then somehow use the relations as polygons.
Thanks for any Reference.