Finiteness of a group given by a presentation

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

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)