# A characterisation of virtually free groups by Gilman R.H., Hermiller S., Holt D.F. By Gilman R.H., Hermiller S., Holt D.F.

We turn out finitely generated staff G is almost unfastened if and provided that there exists a producing set for G and к > zero such that each one k-locally geodesic phrases with recognize to that producing set are geodesic.

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2. The heptagon puzzle. this heptagon’s red (6) and black (0) edges can be. 2 can be. Of course, if the object we are building is to be preserved by the group L, then once we have coloured this edge the colour of the seven corresponding edges is determined by the 7-fold rotational symmetry about our initial face. Clearly this edge cannot itself be red or black as this would give rise to a heptagon with two edges of the same colour. Nor can it be coloured green (5), as the 7-fold symmetry realized by a would then require the third edge at the black/blue (0/1) vertex of face a to be coloured red (6).

We now need a lemma which tells us that a permutation fixing a Steiner system S 5 8 24 cannot fix too many points. 7 A permutation of which preserves a Steiner system S 5 8 24 fixes every point of an octad, and a point not in this octad must be the identity. Proof Let be the permutation, let U = a1 a2 a8 be the octad which is fixed pointwise by , and let b be the fixed point not in U . Let x = b be any other point not in U . We shall show that x must be fixed by and so is the identity. Consider firstly the octad containing a1 a2 a3 b x .

But there are octads which contain a1 but not a2 , and such an octad would be taken by to an octad having seven points in common with itself: a clear contradiction. What then is the action of the stabilizer in M of an octad on the points of that octad? 3 The stabilizer in M of an octad of the Steiner system S 5 8 24 acts as the alternating group A8 on the points of the octad, and has shape 24 A8 . /2 × 759 = 244 823 040. 4 Independent proofs 37 Proof Since 759 is not congruent to 0 modulo seven, we know that elements of order 7 in L must fix octads and act with cycle-shape 1 7 on the points of the octad (indeed, we know that our element aˆ fixes the octad u + ).