# Topological methods to solve equations over groups

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1 Topological methods to solve equations over groups Andreas Thom TU Dresden, Germany July 26, 2016 in Buenos Aires XXI Coloquio Latinoamericano de Álgebra

2 How to solve a polynomial equation?

3 How to solve a polynomial equation? Even though not every non-constant polynomial p(t) Q[t] has a root in Q, there always exists a finite field extension Q K, such that p(t) = 0 can be solved in K,

4 How to solve a polynomial equation? Even though not every non-constant polynomial p(t) Q[t] has a root in Q, there always exists a finite field extension Q K, such that p(t) = 0 can be solved in K, i.e., there exists α K with p(α) = 0.

5 How to solve a polynomial equation? Even though not every non-constant polynomial p(t) Q[t] has a root in Q, there always exists a finite field extension Q K, such that p(t) = 0 can be solved in K, i.e., there exists α K with p(α) = Consider a simple quotient Q[t]/ p(t) K and and convince yourself that Q K. The image α of t will solve the equation p(t) = 0 in K.

6 How to solve a polynomial equation? Even though not every non-constant polynomial p(t) Q[t] has a root in Q, there always exists a finite field extension Q K, such that p(t) = 0 can be solved in K, i.e., there exists α K with p(α) = Consider a simple quotient Q[t]/ p(t) K and and convince yourself that Q K. The image α of t will solve the equation p(t) = 0 in K. 2. Embed Q C, study the continuous map p : C C, and use a topological argument to see that there exists α C, such that p(α) = 0.

7 How to solve a polynomial equation? Even though not every non-constant polynomial p(t) Q[t] has a root in Q, there always exists a finite field extension Q K, such that p(t) = 0 can be solved in K, i.e., there exists α K with p(α) = Consider a simple quotient Q[t]/ p(t) K and and convince yourself that Q K. The image α of t will solve the equation p(t) = 0 in K. 2. Embed Q C, study the continuous map p : C C, and use a topological argument to see that there exists α C, such that p(α) = 0. The second argument was essentially already present in Gauss first proof in 1799.

8 How to solve a polynomial equation? Even though not every non-constant polynomial p(t) Q[t] has a root in Q, there always exists a finite field extension Q K, such that p(t) = 0 can be solved in K, i.e., there exists α K with p(α) = Consider a simple quotient Q[t]/ p(t) K and and convince yourself that Q K. The image α of t will solve the equation p(t) = 0 in K. 2. Embed Q C, study the continuous map p : C C, and use a topological argument to see that there exists α C, such that p(α) = 0. The second argument was essentially already present in Gauss first proof in However, the right language was not developed until 1930.

9 Equations over groups

10 Equations over groups Definition Let Γ be a group and let g 1,..., g n Γ, ε 1,..., ε n Z.

11 Equations over groups Definition Let Γ be a group and let g 1,..., g n Γ, ε 1,..., ε n Z. We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn has a solution in Γ if there exists h Γ such that w(h) = e.

12 Equations over groups Definition Let Γ be a group and let g 1,..., g n Γ, ε 1,..., ε n Z. We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn has a solution in Γ if there exists h Γ such that w(h) = e. The equation has a solution over Γ if there is an extension Γ Λ and there is some h Λ such that w(h) = e in Λ.

13 Equations over groups Definition Let Γ be a group and let g 1,..., g n Γ, ε 1,..., ε n Z. We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn has a solution in Γ if there exists h Γ such that w(h) = e. The equation has a solution over Γ if there is an extension Γ Λ and there is some h Λ such that w(h) = e in Λ. The study of equations like this goes back to: Bernhard H. Neumann, Adjunction of elements to groups, J. London Math. Soc. 18 (1943), 411.

14 Example If a, b Γ, then w(t) = atbt 1 cannot be solved over Γ unless the orders of a and b agree.

15 Example If a, b Γ, then w(t) = atbt 1 cannot be solved over Γ unless the orders of a and b agree. Indeed, if such a t exists, then a 1 = tbt 1.

16 Example If a, b Γ, then w(t) = atbt 1 cannot be solved over Γ unless the orders of a and b agree. Indeed, if such a t exists, then a 1 = tbt 1. Example The equation w(t) = tat 1 ata 1 t 1 a 2 cannot be solved over Z/pZ = a.

17 Example If a, b Γ, then w(t) = atbt 1 cannot be solved over Γ unless the orders of a and b agree. Indeed, if such a t exists, then a 1 = tbt 1. Example The equation w(t) = tat 1 ata 1 t 1 a 2 cannot be solved over Z/pZ = a. Indeed, if w(t) = 1, then a 2 = (tat 1 )a(tat 1 ) 1 and a conjugate of a (namely tat 1 ) would conjugate a to a 2.

18 Example If a, b Γ, then w(t) = atbt 1 cannot be solved over Γ unless the orders of a and b agree. Indeed, if such a t exists, then a 1 = tbt 1. Example The equation w(t) = tat 1 ata 1 t 1 a 2 cannot be solved over Z/pZ = a. Indeed, if w(t) = 1, then a 2 = (tat 1 )a(tat 1 ) 1 and a conjugate of a (namely tat 1 ) would conjugate a to a 2. But the automorphism of Z/pZ which sends 1 to 2 has order dividing p 1 and hence the order is co-prime to p.

19 Definition We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn non-singular if n i=1 ε i 0. is

20 Definition We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn is non-singular if n i=1 ε i 0. It is called non-trivial if it is not conjugate to w(t) = g 1.

21 Definition We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn is non-singular if n i=1 ε i 0. It is called non-trivial if it is not conjugate to w(t) = g 1. Conjecture (Levin) Any non-trivial equation can be solved over Γ, if Γ is torsionfree.

22 Definition We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn is non-singular if n i=1 ε i 0. It is called non-trivial if it is not conjugate to w(t) = g 1. Conjecture (Levin) Any non-trivial equation can be solved over Γ, if Γ is torsionfree. Conjecture (Kervaire-Laudenbach) If w(t) is non-singular, then w(t) has a solution over Γ.

23 Definition We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn is non-singular if n i=1 ε i 0. It is called non-trivial if it is not conjugate to w(t) = g 1. Conjecture (Levin) Any non-trivial equation can be solved over Γ, if Γ is torsionfree. Conjecture (Kervaire-Laudenbach) If w(t) is non-singular, then w(t) has a solution over Γ. Theorem (Klyachko) If Γ is torsionfree and w(t) is non-singular, then w(t) can be solved over Γ. Anton A. Klyachko, A funny property of sphere and equations over groups, Comm. Algebra 21 (1993), no. 7,

24 Definition We say that the equation w(t) = g 1 t ε 1 g 2 t ε 2 g 3... g n t εn is non-singular if n i=1 ε i 0. It is called non-trivial if it is not conjugate to w(t) = g 1. Conjecture (Levin) Any non-trivial equation can be solved over Γ, if Γ is torsionfree. Conjecture (Kervaire-Laudenbach) If w(t) is non-singular, then w(t) has a solution over Γ. Theorem (Klyachko) If Γ is torsionfree and w(t) is non-singular, then w(t) can be solved over Γ. Anton A. Klyachko, A funny property of sphere and equations over groups, Comm. Algebra 21 (1993), no. 7, We will focus on the second conjecture.

25 The algebraic/combinatorial approach Why is this complicated?

26 The algebraic/combinatorial approach Why is this complicated? Just consider: Γ Γ t w(t).

27 The algebraic/combinatorial approach Why is this complicated? Just consider: Γ Γ t w(t). But nobody can show easily that this homomorphism is injective.

28 The algebraic/combinatorial approach Why is this complicated? Just consider: Γ Γ t w(t). But nobody can show easily that this homomorphism is injective. In fact, injectivity is equivalent to existence of a solution over Γ.

29 The algebraic/combinatorial approach Why is this complicated? Just consider: Γ Γ t w(t). But nobody can show easily that this homomorphism is injective. In fact, injectivity is equivalent to existence of a solution over Γ. The Kervaire-Laudenbach conjecture was motivated originally from 3-dimensional topology, where certain geometric operations on knot complements amount to the attachment of an arc and a disc.

30 The algebraic/combinatorial approach Why is this complicated? Just consider: Γ Γ t w(t). But nobody can show easily that this homomorphism is injective. In fact, injectivity is equivalent to existence of a solution over Γ. The Kervaire-Laudenbach conjecture was motivated originally from 3-dimensional topology, where certain geometric operations on knot complements amount to the attachment of an arc and a disc. The resulting effect on fundamental groups is exactly Γ Γ t w(t).

31 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n).

32 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn.

33 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn. Since U(n) is connected, each g i can be moved continuously to 1 n.

34 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn. Since U(n) is connected, each g i can be moved continuously to 1 n. Thus, this map is homotopic to t t i ε i,

35 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn. Since U(n) is connected, each g i can be moved continuously to 1 n. Thus, this map is homotopic to t t i ε i, which has non-trivial degree as a map of topological manifolds.

36 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn. Since U(n) is connected, each g i can be moved continuously to 1 n. Thus, this map is homotopic to t t i ε i, which has non-trivial degree as a map of topological manifolds. Indeed, a generic matrix has exactly d n preimages with d := i ε i.

37 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn. Since U(n) is connected, each g i can be moved continuously to 1 n. Thus, this map is homotopic to t t i ε i, which has non-trivial degree as a map of topological manifolds. Indeed, a generic matrix has exactly d n preimages with d := i ε i. Hence, the map w must be surjective.

38 Topological methods Theorem (Gerstenhaber-Rothaus, 1962) Any non-singular equation in U(n) can be solved in U(n). Proof. Consider the word map w : U(n) U(n), w(t) = g 1 t ε 1... g n t εn. Since U(n) is connected, each g i can be moved continuously to 1 n. Thus, this map is homotopic to t t i ε i, which has non-trivial degree as a map of topological manifolds. Indeed, a generic matrix has exactly d n preimages with d := i ε i. Hence, the map w must be surjective. Each pre-image of 1 n gives a solution of the equation w(t) = 1 n.

39 Corollary Any non-singular equation with coefficients in a finite group Γ can be solved over Γ.

40 Corollary Any non-singular equation with coefficients in a finite group Γ can be solved over Γ. In fact, they can be solved in a finite extension Γ Λ.

41 Corollary Any non-singular equation with coefficients in a finite group Γ can be solved over Γ. In fact, they can be solved in a finite extension Γ Λ. Proof. Embed Γ in U(n), solve the equation there to get a solution u.

42 Corollary Any non-singular equation with coefficients in a finite group Γ can be solved over Γ. In fact, they can be solved in a finite extension Γ Λ. Proof. Embed Γ in U(n), solve the equation there to get a solution u. Now, Λ = Γ, u U(n) is residually finite my Mal cev s theorem. Thus, there exists a finite quotient of Λ which contains Γ.

43 Corollary Any non-singular equation with coefficients in a finite group Γ can be solved over Γ. In fact, they can be solved in a finite extension Γ Λ. Proof. Embed Γ in U(n), solve the equation there to get a solution u. Now, Λ = Γ, u U(n) is residually finite my Mal cev s theorem. Thus, there exists a finite quotient of Λ which contains Γ. The same holds for locally residually finite groups, but the general situation remained unclear back in the 60s.

44 Corollary Any non-singular equation with coefficients in a finite group Γ can be solved over Γ. In fact, they can be solved in a finite extension Γ Λ. Proof. Embed Γ in U(n), solve the equation there to get a solution u. Now, Λ = Γ, u U(n) is residually finite my Mal cev s theorem. Thus, there exists a finite quotient of Λ which contains Γ. The same holds for locally residually finite groups, but the general situation remained unclear back in the 60s.

45 Definition We say that Γ is algebraically closed if any non-singular equation has a solution in Γ.

46 Definition We say that Γ is algebraically closed if any non-singular equation has a solution in Γ. Proposition The class of algebraically closed groups is closed under products and quotients.

47 Definition We say that Γ is algebraically closed if any non-singular equation has a solution in Γ. Proposition The class of algebraically closed groups is closed under products and quotients. Corollary (Pestov) Any group Γ that embeds into an abstract quotient of n U(n) (these are called hyperlinear) satisfies Kervaire s Conjecture.

48 Definition We say that Γ is algebraically closed if any non-singular equation has a solution in Γ. Proposition The class of algebraically closed groups is closed under products and quotients. Corollary (Pestov) Any group Γ that embeds into an abstract quotient of n U(n) (these are called hyperlinear) satisfies Kervaire s Conjecture. Remark Every sofic group can be embedded into a quotient of n U(n).

49 Sofic groups Definition Let Sym(n) be the permutation group on n letters. We set: d(σ, τ) = 1 {i {0,..., n} σ(i) τ(i)} n to be the normalized Hamming distance on permutations σ, τ Sym(n).

50 Sofic groups Definition Let Sym(n) be the permutation group on n letters. We set: d(σ, τ) = 1 {i {0,..., n} σ(i) τ(i)} n to be the normalized Hamming distance on permutations σ, τ Sym(n). Definition A group Γ is called sofic, if for every finite subset F Γ and every ɛ (0, 1) there exists n N and a map φ: Γ Sym(n), such that:

51 Sofic groups Definition Let Sym(n) be the permutation group on n letters. We set: d(σ, τ) = 1 {i {0,..., n} σ(i) τ(i)} n to be the normalized Hamming distance on permutations σ, τ Sym(n). Definition A group Γ is called sofic, if for every finite subset F Γ and every ɛ (0, 1) there exists n N and a map φ: Γ Sym(n), such that: 1. d(φ(gh), φ(g)φ(h)) ɛ, g, h F,

52 Sofic groups Definition Let Sym(n) be the permutation group on n letters. We set: d(σ, τ) = 1 {i {0,..., n} σ(i) τ(i)} n to be the normalized Hamming distance on permutations σ, τ Sym(n). Definition A group Γ is called sofic, if for every finite subset F Γ and every ɛ (0, 1) there exists n N and a map φ: Γ Sym(n), such that: 1. d(φ(gh), φ(g)φ(h)) ɛ, g, h F, 2. d(1 n, φ(g)) 1/2, g F \ {e}.

53 Sofic groups Examples Examples of sofic groups: residually finite groups,

54 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite.

55 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite. amenable groups,

56 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite. amenable groups, inverse and direct limits of sofic groups,

57 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite. amenable groups, inverse and direct limits of sofic groups, free and direct products of sofic groups,

58 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite. amenable groups, inverse and direct limits of sofic groups, free and direct products of sofic groups, subgroups of sofic groups,

59 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite. amenable groups, inverse and direct limits of sofic groups, free and direct products of sofic groups, subgroups of sofic groups, extension with sofic normal subgroup and amenable quotient.

60 Sofic groups Examples Examples of sofic groups: residually finite groups, free groups are residually finite, Theorem (Mal cev): Every finitely generated subgroup of GL n C is residually finite. amenable groups, inverse and direct limits of sofic groups, free and direct products of sofic groups, subgroups of sofic groups, extension with sofic normal subgroup and amenable quotient. Remark There is no group known to be non-sofic.

61 Now, Sym(n) U(n) and one easily sees that a sofic group Γ is a subgroup of the quotient group n U(n), N

62 Now, Sym(n) U(n) and one easily sees that a sofic group Γ is a subgroup of the quotient group n U(n), N where N = (u n) n n 1 U(n) lim n ω n for some suitable ultrafilter ω βn. n δ ij u ij 2 = 0 i,j=1

63 Now, Sym(n) U(n) and one easily sees that a sofic group Γ is a subgroup of the quotient group n U(n), N where N = (u n) n n 1 U(n) lim n ω n for some suitable ultrafilter ω βn. n δ ij u ij 2 = 0 i,j=1 Remark Connes Embedding Conjecture also implies that every group has such an embedding.

64 More Magical realism with sofic groups

65 More Magical realism with sofic groups... uses fantastical and unreal elements. Miracles happen naturally.

66 More Magical realism with sofic groups... uses fantastical and unreal elements. Miracles happen naturally. Conjecture (Kaplansky) Let Γ be a group and k be a field. If a, b kγ satisfy ab = 1, then also ba = 1.

67 More Magical realism with sofic groups... uses fantastical and unreal elements. Miracles happen naturally. Conjecture (Kaplansky) Let Γ be a group and k be a field. If a, b kγ satisfy ab = 1, then also ba = 1. Known to hold if char(k) = 0.

68 More Magical realism with sofic groups... uses fantastical and unreal elements. Miracles happen naturally. Conjecture (Kaplansky) Let Γ be a group and k be a field. If a, b kγ satisfy ab = 1, then also ba = 1. Known to hold if char(k) = 0. Known for any field if Γ is sofic. (Elek-Szabo)

69 More Magical realism with sofic groups... uses fantastical and unreal elements. Miracles happen naturally. Conjecture (Kaplansky) Let Γ be a group and k be a field. If a, b kγ satisfy ab = 1, then also ba = 1. Known to hold if char(k) = 0. Known for any field if Γ is sofic. (Elek-Szabo) Idea: If Γ can be modelled by permutations, then kγ can be modelled by M n (k). Hence, ab = 1 implies ba = 1.

70 More variables

71 More variables Question Can you solve the equation over some group? w(s, t) = g 1 sg 2 tg 3 s 1 g 4 t 1 = 1

72 More variables Question Can you solve the equation over some group? w(s, t) = g 1 sg 2 tg 3 s 1 g 4 t 1 = 1 Consider the augmentation ε: Γ F n F n. An equation w Γ F n is non-singular, if ε(w) 1.

73 More variables Question Can you solve the equation over some group? w(s, t) = g 1 sg 2 tg 3 s 1 g 4 t 1 = 1 Consider the augmentation ε: Γ F n F n. An equation w Γ F n is non-singular, if ε(w) 1. Conjecture If w Γ F n is non-singular, then w has a solution over Γ.

74 More variables Question Can you solve the equation over some group? w(s, t) = g 1 sg 2 tg 3 s 1 g 4 t 1 = 1 Consider the augmentation ε: Γ F n F n. An equation w Γ F n is non-singular, if ε(w) 1. Conjecture If w Γ F n is non-singular, then w has a solution over Γ. Theorem (with Anton Klyachko) If w Γ F 2 satisfies ε(w) [[F 2, F 2 ], F 2 ] and Γ is hyperlinear, then w has a solution over Γ.

75 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p).

76 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p). Instead of degree theory, we are making use of the cohomology ring of SU(p) and PU(p).

77 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p). Instead of degree theory, we are making use of the cohomology ring of SU(p) and PU(p). The key insight is: 1. The commutator map c : PU(p) 2 PU(p) lifts to SU(p).

78 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p). Instead of degree theory, we are making use of the cohomology ring of SU(p) and PU(p). The key insight is: 1. The commutator map c : PU(p) 2 PU(p) lifts to SU(p). (Indeed, the analogous commutator map c : SU(p) 2 SU(p) factorized through PU(p) 2.)

79 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p). Instead of degree theory, we are making use of the cohomology ring of SU(p) and PU(p). The key insight is: 1. The commutator map c : PU(p) 2 PU(p) lifts to SU(p). (Indeed, the analogous commutator map c : SU(p) 2 SU(p) factorized through PU(p) 2.) 2. The generator of the top-dimensional cohomology group H (SU(p), Z/pZ) is mapped non-trivially to H (PU(p) 2, Z/pZ)

80 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p). Instead of degree theory, we are making use of the cohomology ring of SU(p) and PU(p). The key insight is: 1. The commutator map c : PU(p) 2 PU(p) lifts to SU(p). (Indeed, the analogous commutator map c : SU(p) 2 SU(p) factorized through PU(p) 2.) 2. The generator of the top-dimensional cohomology group H (SU(p), Z/pZ) is mapped non-trivially to H (PU(p) 2, Z/pZ) = H (PU(p), Z/pZ) H (PU(p), Z/pZ).

81 Theorem (with Klyachko) Let p be prime. Any w SU(p) F 2 with ε(w) F p 2 [[F 2, F 2 ], F 2 ] can be solved in SU(p). Instead of degree theory, we are making use of the cohomology ring of SU(p) and PU(p). The key insight is: 1. The commutator map c : PU(p) 2 PU(p) lifts to SU(p). (Indeed, the analogous commutator map c : SU(p) 2 SU(p) factorized through PU(p) 2.) 2. The generator of the top-dimensional cohomology group H (SU(p), Z/pZ) is mapped non-trivially to H (PU(p) 2, Z/pZ) = H (PU(p), Z/pZ) H (PU(p), Z/pZ). 3. Thus, c : PU(p) 2 SU(p) is not homotopic to a non-surjective map.

82 Theorem (Borel) with x i = 2i 1 and H (SU(n), Z/pZ) = Λ Z/pZ (x 2, x 3,..., x n )

83 Theorem (Borel) H (SU(n), Z/pZ) = Λ Z/pZ (x 2, x 3,..., x n ) with x i = 2i 1 and (x i ) = x i x i.

84 Theorem (Baum-Browder) Let p be an odd prime number. Then, H (PU(p), Z/pZ) = (Z/pZ)[y]/(y p ) Z Λ Z/pZ (y 1, y 2,..., y p 1 ) with y = 2, y i = 2i 1.

85 Theorem (Baum-Browder) Let p be an odd prime number. Then, H (PU(p), Z/pZ) = (Z/pZ)[y]/(y p ) Z Λ Z/pZ (y 1, y 2,..., y p 1 ) with y = 2, y i = 2i 1. The co-multiplication takes the form (y) = y y, and

86 Theorem (Baum-Browder) Let p be an odd prime number. Then, H (PU(p), Z/pZ) = (Z/pZ)[y]/(y p ) Z Λ Z/pZ (y 1, y 2,..., y p 1 ) with y = 2, y i = 2i 1. The co-multiplication takes the form (y) = y y, and i 1 ( ) j 1 (y i ) = y i y i + y j y i j. i 1 j=1

87 Theorem (Baum-Browder) Let p be an odd prime number. Then, H (PU(p), Z/pZ) = (Z/pZ)[y]/(y p ) Z Λ Z/pZ (y 1, y 2,..., y p 1 ) with y = 2, y i = 2i 1. The co-multiplication takes the form (y) = y y, and i 1 ( ) j 1 (y i ) = y i y i + y j y i j. i 1 j=1 In particular, the co-multiplication is not co-commutative.

88 Larsen s Conjecture For w F 2 \ {1}, what about the equation w(s, t) = g?

89 Larsen s Conjecture For w F 2 \ {1}, what about the equation w(s, t) = g? It can always be solved over Γ, but can it be solved in PU(n) or SU(n)?

90 Larsen s Conjecture For w F 2 \ {1}, what about the equation w(s, t) = g? It can always be solved over Γ, but can it be solved in PU(n) or SU(n)? Conjecture (Larsen) Let w F 2. If n N w, then the word map w : PU(n) 2 PU(n) is surjective.

91 Larsen s Conjecture For w F 2 \ {1}, what about the equation w(s, t) = g? It can always be solved over Γ, but can it be solved in PU(n) or SU(n)? Conjecture (Larsen) Let w F 2. If n N w, then the word map w : PU(n) 2 PU(n) is surjective. Theorem (with Abdul Elkasapy) If w [[F 2, F 2 ], [F 2, F 2 ]] and n is not divisible a prime in some finite set P w, then Larsen s Conjecture holds for w.

92 Larsen s Conjecture For w F 2 \ {1}, what about the equation w(s, t) = g? It can always be solved over Γ, but can it be solved in PU(n) or SU(n)? Conjecture (Larsen) Let w F 2. If n N w, then the word map w : PU(n) 2 PU(n) is surjective. Theorem (with Abdul Elkasapy) If w [[F 2, F 2 ], [F 2, F 2 ]] and n is not divisible a prime in some finite set P w, then Larsen s Conjecture holds for w. Corollary Engel words w(s, t) = [...[[s, t], t],..., t] are always surjective on groups PU(n).

93 Remark Maybe, for fixed w F 2 \ {1} and n large enough, w : PU(n) 2 PU(n) is not even homotopic to a non-surjective map?

94 Remark Maybe, for fixed w F 2 \ {1} and n large enough, w : PU(n) 2 PU(n) is not even homotopic to a non-surjective map? Maybe at least not homotopically trivial?

95 Remark Maybe, for fixed w F 2 \ {1} and n large enough, w : PU(n) 2 PU(n) is not even homotopic to a non-surjective map? Maybe at least not homotopically trivial? Theorem For any n N, ε > 0, there exists a word w F 2 \ {1} such that w(u, v) 1 n ε, u, v U(n). This solved a longstanding open problem in non-commutative harmonic analysis in the negative.

96 Thank you for your attention!

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