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Groups, ends and trees: exercises I
Michele Triestino 2015-07-21
Exercise 1. Show that SL(n, Z) is generated by the transvections: these are the matrices with 1's on the diagonal and one other entry equal to 1 and all the others 0. Exercise 2. The elements of the lamplighter group Z2 Z = Z can be represented by all the Z Z2 possible configurations of an infinite line of lamps, which can be ON or OFF, with only finitely many lamps ON, with a lamplighter sitting on some marked site. Show that the group is generated by two elements which can be described by the possible actions of the lamplighter: he can switch the light at the site where he seats ON/OFF (which corresponds to an element of order 2), and he can move one step right/left. Is this group finitely presented? Exercise 3. Show that (Q, +) is not finitely generated. Exercise 4. Show that the fundamental group of a bouquet of n circles is the free group over an alphabet S of size n: #S = n. More generally, what is the fundamental group of a finite graph? Exercise 5. Show that the free group F(S) satisfies the following universal property: for any function : S - G from S to a group G, there exists a unique group homomorphism : F(S) - G such that (s) = (s) for any s S. Exercise 6. Show that two free groups F(S) and F(S ) are isomorphic if and only if S and S have the same cardinality. Exercise 7. Prove that the quotient of a finitely generated group is finitely generated. Exercise 8. Let N and K be two finitely generated groups, with N subgroup of a group G, such that G/N = K. Then G is finitely generated. If N and K are moreover finitely presented, so is G. Exercise 9. Let G = s1 , s2 | r be a group with two generators and one relation. Write r = si 1 · · · si n , n 1 with i j = 1, 2 and j = ±1. Let X0 be the bouquet of two circles 1 and 2 , and choose an orientation for both of them. In 1 (X0 ), the homotopy classes s1 = [1 ] and s2 = [2 ] generate the free group F(s1 , s2 ). Then we take a disc D of dimension 2 and cut the boundary D into n labelled and oriented intervals I j , according to the relation r = si 1 · · · si n : going cyclically around the boundary, we read n 1 the label si j on I j , and the orientation is positive if j = 1, negative otherwise. Consider a continuous map : D - X0 , satisfying the condition that (I j ) = i j and the restriction of to the interior of I j is injective and is orientation preserving if j = 1, and orientation reversing otherwise. We define the space X1 to be the topological space obtained from the disjoint union X0 D, and identifying every point p D with the image (p) X0 . Show that the fundamental group of X1 is isomorphic to the group G. 1