# Finiteness of a group given by a presentation

Let $G$ be a group generated by

"s1", "s2", "s3", "s4", "s12", "s23", "s34", "s123", "s234", "s1234"


with relations

[ s1^2, s2^2, s3^2, s4^2,
s12^2, s23^2, s34^2,
s123^2, s234^2,
s1234^2,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4,
s1*s3*s1*s3, s2*s4*s2*s4, s1*s4*s1*s4,
s1*s12*s2*s12,
s4*s34*s3*s34,
s3*s123*s23*s12*s3*s123*s23*s12,
s123*s12*s4*s34*s4*s12*s2,
s23*s1234*s12*s3*s4*s3*s12,
s23*s1234*s34*s1*s2*s1*s34 ]

# s1       s2       s3       s4
# o--------o--------o--------o
#
#    non-standard joining relations
#
#     s12      s23       s34
#     o        o         o
#
#    non-standard joining relations
#
#         s123     s234
#         o        o
#
#    non-standard joining relations
#
#             s1234
#             o


All the above equal to the identity $e$ of $G$.

Is there some function in SageMath which could check if $G$ is finite or not? Thank you very much.

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G.is_finite() ?

( 2022-06-14 18:25:05 +0200 )edit

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Too long for a comment: the "obvious" way to do it fails, at least for me:

sage: G.<s1, s2, s3, s4, s12, s23, s34, s123, s234, s1234> = FreeGroup()
sage: H = G / [s1^2, s2^2, s3^2, s4^2, s12^2, s23^2, s34^2, s123^2, s234^2, s1234^2, s1*s2*s1*s2
....: *s1*s2, s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4, s1*s3*s1*s3, s2*s4*s2*s4, s1*s4*s1*s4,s1*s12
....: *s2*s12, s4*s34*s3*s34, s3*s123*s23*s12*s3*s123*s23*s12, s123*s12*s4*s34*s4*s12*s2, s23*s1
....: 234*s12*s3*s4*s3*s12,s23*s1234*s34*s1*s2*s1*s34]
sage: H.is_finite()
#I  Coset table calculation failed -- trying with bigger table limit
#I  Coset table calculation failed -- trying with bigger table limit
#I  Coset table calculation failed -- trying with bigger table limit
#I  Coset table calculation failed -- trying with bigger table limit
#I  Coset table calculation failed -- trying with bigger table limit
#I  Coset table calculation failed -- trying with bigger table limit
---------------------------------------------------------------------------
GAPError                                  Traceback (most recent call last)

[snip]

GAPError: Error, reached the pre-set memory limit
(change it with the -o command line option)

more

1
( 2022-06-14 19:03:37 +0200 )edit

Thanks for the tip. Increasing the memory leads to longer delays before warnings start printing, but still the same warning. I went up to 48G and still no success.

( 2022-06-15 03:44:15 +0200 )edit

@John, thank you very much!

( 2022-06-15 08:30:04 +0200 )edit

Apparently the group is infinite as it contains a free subgroup generated by s234*s1234. I don't have a proof, but here is a quick check that the first 100 powers of this element are irreducible:

sage: F.<s1, s2, s3, s4, s12, s23, s34, s123, s234, s1234> = FreeGroup()
sage: relators = [s1^2, s2^2, s3^2, s4^2,
....:             s12^2, s23^2, s34^2, s123^2, s234^2, s1234^2,
....:             (s1*s2)^3, (s2*s3)^3, (s3*s4)^3,
....:             (s1*s3)^2, (s2*s4)^2, (s1*s4)^2,
....:             s1*s12*s2*s12, s4*s34*s3*s34,
....:             (s3*s123*s23*s12)^2, s123*s12*s4*s34*s4*s12*s2,
....:             s23*s1234*s12*s3*s4*s3*s12, s23*s1234*s34*s1*s2*s1*s34]
sage: G = F / relators
sage: rw = G.rewriting_system()
sage: g = s234*s1234
sage: rw.reduce(g^99)
(s234*s1234)^99

more

@max, thank you very much!

( 2022-06-15 08:29:49 +0200 )edit