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A359797 Cogrowth sequence of the lamplighter group Z_2 wr Z where wr denotes the wreath product.

1, 3, 15, 87, 547, 3623, 24885, 175591, 1265187, 9271167, 68894785, 518053231, 3935274277, 30158804835, 232930956175, 1811476156847, 14174669041427, 111532445963367, 882004732285473, 7006931317108119, 55899039962599777, 447666261592033123

Offset: 0

Andrew Elvey Price, Jan 13 2023

a(n) is the number of words of length 2n in the letters a,t,t^(-1) that equal the identity of the lamplighter group Z_2 wr Z = .

Walks on this group can be seen as operations on an infinite tape of 0's and 1's where each step is either a right shift, left shift or toggles the current element. a(n) is then the number of sequences of 2n such moves which return the tape to the initial position.

Andrew Elvey Price, Table of n, a(n) for n = 0..500
Andrew Elvey Price and A. J. Guttmann, Numerical studies of Thompson's group F and related groups, arXiv:1706.07571 [math.GR], 2017.
D. Revelle, Heat kernel asymptotics on the lamplighter group, Electronic Communications in Probability 8 (2003), 142-154.
Wikipedia, Lamplighter group

Spherical growth sequence for this group is A288348.

Cf. A359798.

A359798 Cogrowth sequence of the group Z wr Z where wr denotes the wreath product.

Original entry on oeis.org

1, 4, 28, 232, 2108, 20384, 206392, 2165720, 23385340, 258532216, 2915343808, 33437862352, 389230520888, 4590271681064, 54767161155000, 660307913374352, 8036973478493436, 98672644594401736, 1221090110502080440, 15222093531642444504

Offset: 0

Andrew Elvey Price, Jan 13 2023

nonn

a(n) is the number of words of length 2n in the letters a,a^(-1),t,t^(-1) that equal the identity of the group Z wr Z = .

Andrew Elvey Price, Table of n, a(n) for n = 0..500
Andrew Elvey Price and A. J. Guttmann, Numerical studies of Thompson's group F and related groups, arXiv:1706.07571 [math.GR], 2017.
C. Pittet and L. Saloff-Coste, On random walks on wreath products, The annals of probability, 30 No. 2 (2002), 948-977.
Wikipedia, Wreath product

Related cogrowth sequences: A359797, A359705. Spherical growth sequence for this group is A294782.

A359705 Cogrowth sequence of the Brin-Navas group B.

Original entry on oeis.org

1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275692, 211825600, 2240855128, 23952786400, 258287602744, 2806152315048, 30686462795856, 337490492639512, 3730522624066540, 41422293291178872, 461802091590831904, 5167329622166765872

Offset: 0

Andrew Elvey Price, Jan 11 2023

This is the number of words of length 2n equal to the identity in the letters t, t^{-1}, a, a^{-1} which generate the Brin-Navas group B.

			For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.

Andrew Elvey Price, Table of n, a(n) for n = 0..126
Andrew Elvey Price and A. J. Guttmann, Numerical studies of Thompson's group F and related groups, arXiv:1706.07571 [math.GR], 2017.

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A359797 Cogrowth sequence of the lamplighter group Z_2 wr Z where wr denotes the wreath product.

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A359798 Cogrowth sequence of the group Z wr Z where wr denotes the wreath product.

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A359705 Cogrowth sequence of the Brin-Navas group B.

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