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 2 t for 0 ≤ t ≤ 1 4 t + 1 4 for 1 4 ≤ t ≤ 1 x 0 = 2 1 2 t + 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 2 ) n for some positive n . Then: with each subinterval of length ( 1 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 = � x 0 , x 1 , x 2 , ..... | x i x j = x j x i +1 for all i > j � . ◮ F ab ∼ = Z 2 . ◮ F is finitely presented (actually, F is of type F ∞ (Brown-Geoghegan)). Brita Nucinkis Irrational slope Thompson’s groups
Generalisations of F ◮ F n = 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
F τ √ 5 − 1 Let τ be the small Golden Ratio, i.e. τ = . Hence 2 τ 2 + τ = 1 . We define F τ = 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 F τ . Theorem (Cleary 2000) Every element of F τ 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 F τ Brita Nucinkis Irrational slope Thompson’s groups
Properties of F τ ◮ F τ is torsion-free ◮ cdF τ = ∞ ◮ F τ is of type F ∞ (Cleary 2000) ◮ (BNR) An explicit presentation is given as follows: F τ = � x 0 , y 0 , x 1 , y 1 , .... | a i b j = b j a i +1 , a , b ∈ { x , y } , i > j , y 2 i = x i x i +1 � ◮ (BNR) Every element in F τ can be represented by a tree-pair ( T 1 , T 2 ) satisfying the following: 1. T 2 has no y -carets, 2. y -carets in T 1 have no left children. = Z 2 × Z / 2 Z . ◮ (BNR) ( F τ ) ab ∼ 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 + 2 X = 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 only 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 F τ . Brita Nucinkis Irrational slope Thompson’s groups
V and V τ 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 V τ = 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
V τ is not simple Theorem (BNR 2020) V τ is finitely presented and has a simple subgroup of index 2 . Remark Using, by now, standard methods one can also show that V τ is of type F ∞ . (see e.g. Stein, Fluch et.al., Mart´ ınez-P´ erez-Matucci-N) Brita Nucinkis Irrational slope Thompson’s groups
V τ is not simple - Steps in the proof ◮ Lemma 1. The parity of y -carets is independent of the choice of tree-pair. Hence: V τ ։ Z / 2 Z . Let K denote the kernel of this map. ◮ Lemma 2. Every element in K can be represented by a tree pair ( T 1 , π, T 2 ) where T 2 has no y -carets and T 1 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
V τ 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
V η with a normal subgroup of index 4 Let η be the real root in [0 , 1] of X 2 + 2 X = 1. Then there is a normal subgroup V + η of index 2 in V η given by the even permutations, analogously to Higman’s argument for V 3 . But we also have an analogue to K of index 2 in V η . This follows from using J. Brown’s presentation for V η . 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 ]. Illinois J. Math., 44(3):453–464, 2000. 2 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
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