Introduction How does the map look like? Transport network on the map Conclusion of the proof Commutators cannot be proper powers in metric small-cancellation torsion-free groups Liza Frenkel, Moscow State University based on joint result with A. A. Klyachko May, 28, 2013 Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Commutators Vs Proper powers It is well known that Fact (Sch¨ utzenberger, ’59). A nontrivial commutator cannot be a proper power in a free group. In free products, the situation is more complicated but also completely studied. Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Commutators Vs Proper powers Theorem (Comerford, Edmunds, G. Rosenberger, ’94). Let G = ∗ i ∈ I G i , the free product of nontrivial free factors G i . If V , X , Y ∈ G and V m = X − 1 Y − 1 XY = [ X , Y ] for some m ≤ 2, then either (1.1) V ∈ W − 1 G i W for some W ∈ G , i ∈ I , and V m is a commutator in W − 1 G i W ; or (1.2) m is even, V = AB with A 2 = B 2 = 1, and V m = [ A , B ( AB ) ( m − 2) / 2 ]; or (1.3) m is odd, V = AC − 1 AC with A 2 = 1, and V m = [ A , C ( AC − 1 AC ) ( m − 1) / 2 ]; or (1.4) m = 6, V = AB with A 2 = B 3 = 1, and V 6 = [ B − 1 ABA , B ( AB ) 2 ]; or (1.5) m = 3, V = AB with A 3 = B 3 = 1, and V 3 = [ BA − 1 , BAB ]; or (1.6) m = 2, V = AB with A 2 = 1 and B − 1 = C − 1 BC for some C ∈ G , and V 2 = [ C − 1 A , B ]; or (1.7) m = 4, V 2 = ABC with A 2 = B 2 = C 2 = 1, and V 4 = [ BA , BC ]. Some partial results are known about amalgamated products (Fine, Rosenberger A., Rosenberger G.) Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Main result Theorem (F., Klyachko, ’12). If a torsion-free group satisfies the C ′ ( λ ) small cancellation condition for sufficiently small λ , then no nontrivial commutator can be a proper power in this group. Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Definitions A presentation � X | R � satisfies the C ′ ( λ ) small cancellation condition if it is symmetrised (i.e. R is closed under taking cyclic permutations and inverses) and, for any two different relators r 1 , r 2 ∈ R , the length of their common initial segment is less than λ | r 1 | . Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Intuition Example. Let us show that a nontrivial commutator cannot be a cube in the free group F ( a , b ): Theorem (Wicks, ’62). A cyclically reduced word in a free group F is a commutator if and only if a cyclic permutation of this word is graphically equal to xyzx − 1 y − 1 z − 1 , where x , y , and z are some reduced words. We have a graphical equality of the form xyzx − 1 y − 1 z − 1 = www . Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Intuition Geometrically: There is a graph on a torus, and all vertices have degree two, except two vertices of degree three, or except one vertex of degree four (the case where one of the words x , y , or z is empty). The edges of the graph are directed and labeled by letters a and b . The complement to this graph is homeomorphic to a disk. � y ❡ � � ✠ � � � x ✛ ✛ ❡ ❡ � z � � � Three cars move counterclockwise along the boundary of this disk with a constant speed of one edge per minute. This motion is periodic with period | w | , i.e. each | w | minutes the cars cyclically interchange (each car “reads” the word w in | w | minutes). Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Intuition According to the car-crash lemma (!), every such periodic motion on a torus leads to a collision. This collision can occur only at a vertex: If a car is driving along an edge labeled, say a , in the positive direction, then all remaining cars are also driving edges labeled a in the positive direction at this moment (so, they cannot collide). A collision at a vertex contradicts the irreducibility of the boundary label of the disk (i.e. of the word xyzx − 1 y − 1 z − 1 ). Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Intuition The case of a non-free group: Instead of the simple graph as above, we have a van Kampen diagram. However, for small cancellation groups, this diagram turns out to be very thin and similar to this graph. This allows us to apply “automobile technique”, although the arguments become much more complicated. Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Maps and metric small cancellation conditions A map on a closed surface is a finite graph on this surface that divides the surface into simply connected domains, called cells. Some cells are distinguished and called holes; the remaining cells are called interior. The boundary δ Γ and the perimeter | δ Γ | of a cell Γ are defined naturally. A piece is a simple path of positive length in the map that joins two vertices (possibly coinciding) of degree other than two, and does not pass through other vertices of degree other than two. Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Maps and metric small cancellation conditions We say that a map without vertices of degree one satisfies C ′ ( λ ) condition, where λ is a nonnegative real number if the length of any common piece of the boundaries of two interior cells Γ 1 and Γ 2 is less than λ | δ Γ 1 | , and the length of any piece that separates an interior cell Γ and a hole is less than ( 1 2 + λ ) | δ Γ | . In what follows, we consider only maps on a torus with one hole. Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Our principles We widely use the lowest parameter principle (introduced by Olshanskiy, ’89): there are fixed small positive numbers λ ≪ λ 1 ≪ λ 2 ≪ λ 3 ≪ λ 4 ≪ λ 5 ≪ λ 6 ≪ λ 7 ≪ 1 and each time an inequality of the form, e.g., 2013 λ 2 < λ 3 arises, we conclude that it is automatically fulfilled by the choice of λ i . Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Structure of the map Lemma 1. If λ ≪ 1, then the boundary of each interior cell of a C ′ ( λ )-map on the torus with a hole contains at least two pieces lying on the hole boundary. We call an interior cell ordinary if its boundary has two common pieces with the hole boundary. An interior cell is called 1-special if its boundary has three common pieces with the hole. An interior cell is 2-special if its boundary has four common pieces with the boundary of the hole. Lemma 2. If λ ≪ 1, then each interior cell of a C ′ ( λ )-map on a torus with one hole is either ordinary, 1-special, or 2-special. The number of all special cells is at most two; if the map has a 2-special cell, then there are no other special cells. All vertices not lying on the hole boundary, except may be two, have degree two, and all vertices lying on the hole boundary, except may be two, have degree two or three. Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
Introduction How does the map look like? Transport network on the map Conclusion of the proof Structure of the map Fact (Combinatorial Gauss-Bonnet formula). If each corner c of a map on a closed surface S is assigned a number ν ( c ) (called the weight or the value of the corner c ) , then � � � K ( v ) + K ( D ) + K ( e ) = 2 χ ( S ) . v e D Here, the summations are over all vertices v and all cells D of the map, and the values K ( v ), K ( D ), and K ( e ), called the curvatures of the corresponding vertex, cell, and edge, are defined by the formulae � � K ( v ) = 2 − ν ( c ) , K ( D ) = 2 − (1 − ν ( c )) , K ( e ) = 0 , c c where the first sum is over all corners at the vertex v , and the second sum is over all corners of the cell D . Liza Frenkel, Moscow State University Commutators cannot be proper powers in metric small-cancellation torsion-f
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