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Extra info for 2-Groups which contain exactly three involutions
For α > 0, the most often used C (α) condition states that the length of any piece is less than α times the infimum of the lengths of the relators on which it appears. For an integer p, the C(p) condition holds if no relator is the union of less than p pieces. The B(2p) condition holds if the union of p consecutive pieces always makes less than half a relator. We have the implications C (1/2p) ⇒ B(2p) ⇒ C(2p + 1). The T (p) small cancellation condition is totally irrelevant for groups with lots of relators.
Actually the technique used in [AC04] allows to embed subdivisions of lots of finite graphs into the Cayley graph of a small-density random group. e. Growth exponent. The growth exponent of a group presentation G = a1 , . . , am | R measures the rate of growth of balls in the group. Let BL be the set of elements of the group G which can be written ±1 as a word of length at most L in the generators a±1 1 , . . , am . If G is the L free group Fm , the number of elements of BL is 1 + k=1 (2m)(2m − 1)L−1 which is the number of elements at distance at most L from the origin in the valency-2m regular tree.
Once Theorem 19 is known the result is relatively simple. Indeed, a theorem of Magnus ([LS77], Prop. 8) implies that if two elements of the free group generate the same normal subgroups then they are the same up to conjugation and inversion. If two generic one-relator groups are isomorphic, then Theorem 19 implies that after applying some automorphism of the free group, the two relators have the same normal closure, and thus are essentially the same by Magnus’ Theorem. So the isomorphism problem for generic one-relator groups reduces to the problem of knowing when an element in the free group is the image of another under some automorphism of the free group.