Irrational slope Thompsons groups Brita Nucinkis joint with J. - - PowerPoint PPT Presentation

Irrational slope Thompson’s groups

Brita Nucinkis joint with J. Burillo and L. Reeves

Brita Nucinkis Irrational slope Thompson’s groups

The Bieri-Strebel Groups

Definition

(Bieri-Strebel 1980s) Let I ⊆ R be an interval, A a subring of R, and Λ ≤ A∗ be a subgroup of the group of units of A. Then we denote by G(I, A, Λ) the group of piecewise linear orientation preserving homeomorphisms of I with break-points in A and slopes in Λ.

Remark

◮ The original definition is slightly more general, but this one is more suited

to our purpose.

◮ Throughout this talk we will have I = [0, 1].

Brita Nucinkis Irrational slope Thompson’s groups

Thompson’s group F

With this notation we have F = G([0, 1], Z[1 2], 2).

Brita Nucinkis Irrational slope Thompson’s groups

An element of F

Consider the element x0 =      2t for 0 ≤ t ≤ 1

4

t + 1

4 for 1 4 ≤ t ≤ 1 2 1 2t + 1 2 for 1 2 ≤ t ≤ 1

Brita Nucinkis Irrational slope Thompson’s groups

Describing elements of F

Successively partition [0, 1] by halving each subinterval to obtain a partition with each subinterval of length ( 1

2)n for some positive n. Then:

A pair of such partitions having the same number of subintervals determines an element of F, and each element of F can be obtained that way.

Brita Nucinkis Irrational slope Thompson’s groups

Elements of F via tree-pairs

Every element of F can be represented by a pair of finite binary rooted trees with the same number of leaves.

Brita Nucinkis Irrational slope Thompson’s groups

Properties of F

◮ F is torsion-free ◮ F contains arbitrarily large free abelian groups (and hence cdF = ∞). ◮ F = x0, x1, x2, ..... | xixj = xjxi+1 for all i > j. ◮ Fab ∼

= Z2.

◮ F is finitely presented (actually, F is of type F∞ (Brown-Geoghegan)).

Brita Nucinkis Irrational slope Thompson’s groups

Generalisations of F

◮ Fn = G([0, 1], Z[ 1 n], n); ◮ Groups with non-cyclic slope groups, for example

F{2,3} = G([0, 1], Z[ 1

6], 2, 3), see (Melanie Stein) ◮ All these groups share most of the properties of F, i.e. they are of infinite

cohomological dimension and of type F∞ (Brown, Stein).

Brita Nucinkis Irrational slope Thompson’s groups

Fτ

Let τ be the small Golden Ratio, i.e. τ =

√ 5−1 2

. Hence τ 2 + τ = 1. We define Fτ = G([0, 1], Z[τ], τ).

Brita Nucinkis Irrational slope Thompson’s groups

Subdivisions

[0,1] can be subdivided in two ways into a subintervals of length τ and τ 2 : [0, 1] = [0, τ] ∪ [τ, 1] and [0, 1] = [0, τ 2] ∪ [τ 2, 1]. Repeated subdivision (each involving a choice) gives a partition each of whose subintervals has length a power of τ.. A pair of such partitions having the same number of subintervals determines an element of Fτ.

Theorem

(Cleary 2000) Every element of Fτ can be obtained this way.

Remark

The crucial step towards the theorem is to show that every element of Z[τ] ∩ [0, 1] is a breakpoint in a subdivision as above.

Brita Nucinkis Irrational slope Thompson’s groups

Tree pairs

Brita Nucinkis Irrational slope Thompson’s groups

Generators of Fτ

Brita Nucinkis Irrational slope Thompson’s groups

Properties of Fτ

◮ Fτ is torsion-free ◮ cdFτ = ∞ ◮ Fτ is of type F∞ (Cleary 2000) ◮ (BNR) An explicit presentation is given as follows:

Fτ = x0, y0, x1, y1, .... | aibj = bjai+1 , a, b ∈ {x, y} , i > j, y 2

i = xixi+1 ◮ (BNR) Every element in Fτ can be represented by a tree-pair (T1, T2)

satisfying the following:

• 1. T2 has no y-carets,
• 2. y-carets in T1 have no left children.

◮ (BNR) (Fτ)ab ∼

= Z2 × Z/2Z.

Brita Nucinkis Irrational slope Thompson’s groups

Aside

The question arises whether we can do similar constructions for other algebraic

• numbers. Cleary had results for the real solution in [0, 1] to X 2 + 2X = 1.

The crucial result is the the following:

Theorem

(N. Winstone) Let η be the unique real solution to aX 2 + bX = 1 in [0, 1]. Then every x ∈ [0, 1] ∩ Z[η] is a breakpoint of a η-regular subdivision if and

• nly if a ≤ b.

Furthermore (J. Brown, N. Winstone): if a ≤ b we can represent elements of G([0, 1], Z[η], η) by tree-pair diagrams analogous to Fτ.

Brita Nucinkis Irrational slope Thompson’s groups

V and Vτ

Definition

Let I = (0, 1], A a subring of R, and Λ ≤ A∗ be a subgroup of the group of units of A. Then we denote by V (I, A, Λ) the group of piecewise linear orientation preserving left-continuous maps of I with break-points in A and slopes in Λ. Hence V = V (I, Z[ 1

2], 2 and Vτ = V (I, Z[τ], τ.

Brita Nucinkis Irrational slope Thompson’s groups

Properties of V

◮ V is finitely presented and simple (Higman). ◮ V is of type F∞ (K. Brown). ◮ Every finite group can be embedded into V .

Brita Nucinkis Irrational slope Thompson’s groups

Vτ is not simple

Theorem

(BNR 2020) Vτ is finitely presented and has a simple subgroup of index 2.

Remark

Using, by now, standard methods one can also show that Vτ is of type F∞. (see e.g. Stein, Fluch et.al., Mart´ ınez-P´ erez-Matucci-N)

Brita Nucinkis Irrational slope Thompson’s groups

Vτ is not simple - Steps in the proof

◮ Lemma 1. The parity of y-carets is independent of the choice of tree-pair.

Hence: Vτ ։ Z/2Z. Let K denote the kernel of this map.

◮ Lemma 2. Every element in K can be represented by a tree pair

(T1, π, T2) where T2 has no y-carets and T1 has an even number of y-carets each of which has no children. Lemma 3. K is generated by permutations.

Brita Nucinkis Irrational slope Thompson’s groups

Vτ is not simple -steps in the proof

The following 2 results are adapted from Higman’s original proof and Brin’s version of it, that V is simple.

◮ Lemma 4. Any normal subgroup of K contains a proper transposition. ◮ Lemma 5. Any two proper transpositions are conjugate.

This now proves that K is simple

Brita Nucinkis Irrational slope Thompson’s groups

Vη with a normal subgroup of index 4

Let η be the real root in [0, 1] of X 2 + 2X = 1. Then there is a normal subgroup V +

η of index 2 in Vη given by the even permutations, analogously to

Higman’s argument for V3. But we also have an analogue to K of index 2 in Vη. This follows from using J. Brown’s presentation for Vη.

Brita Nucinkis Irrational slope Thompson’s groups

Bibliography

• M. G. Brin, Higher dimensional Thompson groups, Geom. Dedicata 108

(2004), 163–192.

• J. Brown, A Class of PL-Homeomorphism Groups with Irrational Slopes, M.Sc

Thesis, University of Melbourne, 2018.

• K. S. Brown.

Finiteness properties of groups. Journal of Pure and Applied Algebra, 44, 45–75, 1987.

• J. Burillo, B. Nucinkis, and L. Reeves, An irrational slope Thompson’s group,

preprint, arXiv:1806.00108, 2018.

• J. Burillo, B. Nucinkis, and L. Reeves, An irrational slope Thompson’s group II,

preprint, in preparation, 2020.

• S. Cleary, Groups of piecewise-linear homeomorphisms with irrational slopes.

Rocky Mountain J. Math. 25 (1995), no. 3, 935–955.

• S. Cleary, Regular subdivision in [ 1+

√ 5 2

]. Illinois J. Math., 44(3):453–464, 2000.

Brita Nucinkis Irrational slope Thompson’s groups

Bibliography

• M. Fluch, M. Schwandt, S. Witzel, and M. C. B. Zaremsky.

The Brin-Thompson groups sV are of type F∞. Pacific J. Math. 266 (2013), no. 2, 283–295.

• G. Higman, Finitely presented infinite simple groups, Notes on Pure

Mathematics, Volume 8 (Australian National University, Canberra, 1974).

• C. Mart´

ınez-P´ erez, F. Matucci and B. E. A. Nucinkis. Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group V , to appear in Forum Mathematicum, http://arxiv.org/abs/1309.7858

• C. Mart´

ınez-P´ erez, and B. E. A. Nucinkis. Bredon cohomological finiteness conditions for generalisations of Thompson’s groups, Groups Geom. Dyn. 7 (2013), 931–959,

• M. Stein.

Groups of piecewise linear homeomorphisms.

• Trans. Amer. Math. Soc., 332(2):477–514, 1992.
• N. Winstone,

Irrational slope Thompson groups Ph.D thesis, Royal Holloway, University of London, in preparation.

Brita Nucinkis Irrational slope Thompson’s groups