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Groups, ends and trees: exercises II
Michele Triestino 2015-07-22

Exercise 1. Give examples of groups which are not the fundamental group of a closed surface. Exercise 2. Draw a picture of the Cayley graph of Z3 Z4 . Exercise 3. Let G = S be a group generated by the set S. For any x G, define its word length x to be the length of the shortest word in the alphabet S S-1 that represents the element x: x
S S

= min{n | s1 , . . . , sn S S-1 s.t. x = s1 § § § sn }.

Show that the function dS : G Ѕ G - G defined by dS (x, y) = x-1 y S is a distance on G and coincides with the graph distance on the Cayley graph of G with respect to generating set S. Exercise 4. Let G = S be a finitely generated group and let (G, S) be its Cayley graph, equipped with the graph distance dS . Show that the multiplication in G defines a distance-preserving transitive action of G on (G, S): for any two vertices x, y (G, S), and for any G, dS ( .x, . y) = dS (x, y). Show that the quotient space for this action is homeomorphic to a bouquet of #S circles. Exercise 5. Let S and S be two different finite generating sets of a group G. Show that the Cayley graphs (G, S) and (G, S) are quasi-isometric. Exercise 6. Show that the Cayley graph of a finite group is quasi-isometric to a point. More generally, every compact metric space is quasi-isometric to a point. Exercise 7. Let H G be a finite index subgroup of G. Show that H is finitely generated if and only if G is. Suppose that G = S and H = S are finitely generated. Show that the Cayley graphs (G, S) and (H, S ) are quasi-isometric. Exercise 8. Show that quasi-isometry is an equivalence relation.

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2 Exercise 9. Let (M, d ) be a compact Riemannian manifold. a) Show that it is possible to define a metric d on the universal cover M of M in such a way that the monodromy action of the fundamental group 1 (M) on (M, d ) preserves the distances: 1 (M) is a subgroup of the group of isometries of (M, d ). b) Let U M be a fundamental domain for the monodromy action: U = (M), where : M - M is any continuous section of the covering projection : M - M, that is is injective and = id|M . c) Show that the set S = {s 1 (M) | s is finite. d) Show that пп inf{d (U, xU) | x 1 (M) - (S {id})} =: 2d пп is a minimum and strictly positive. Moreover, if d (U, xU) < 2d then x S {id}. e) Let us fix p M. For any x 1 (M), denote by [p, x.p] a geodesic path from p to x.p. Write k := and let us take points y0 = p, y1 , . . . , yk , y on the geodesic curve [p, x.p], such that d ( yi , y п hi 1 (M) such that yi hi U. Then h-1 hi+1 S. i
i+1 k +1

п п id and sU U

}

d (p, x.p) d = x.p

) d for any i = 0, . . . , k. For any i, consider

This implies that S generates the fundamental group of M: 1 (M) = S . f) The set of points 1 (M).p := {x.p | x 1 (M)} is discrete in M and there exists D > 0 such that the neighbourhood of radius 2D, B2D (1 (M).p) =
x1 (M)

B2D (x.p),

is the whole manifold M. g) For any x 1 (M), let m := x
S

= dS (id, x). Show that

1 1 d (p, x.p) m k + 1 d (p, x.p) + 1, 2D d and hence (1 (M), dS ) and (M, d ) are quasi-isometric. This result is a theorem proved indepen denlty by Svarcz and Milnor.