Lec 11, Limit Groups, R-trees Theorem (Kharl, Mias,razzzz). Suppose H is not free. Then there is a finite set S = fs : H Hsg of proper epimorphisms such that: • for all h 2 Hom(H; F); there exists a 2 Aut(H) such that h ◦ a factors through S. a .... H ................................................................................................................... H . s . h . ....... ....... ... ... ... Hs ................................................................................................................ F • We will refine this statement and discuss a proof. Real trees • Areal tree ( T; dT )is a metric space such that between any two points t; t0 2 T , there is a unique arc∗ from t to t0 and this arc is the image of an isometric embedding of an interval. Example. A finite simplicial real tree is a finite tree with each edge identified with an interval. Example. A countable increasing union of finite simplicial real trees. Example. 0-hyperbolic spaces embed into real trees. ∗the image of an embedding σ : [x; x0] ! T with σ(x) = t and σ(x0) = t0 1 Isometries of real trees • An isometry η of a real tree T either elliptic or hyperbolic. • Elliptic η fixes a point of T . The axis of η is F ix(η). • Hyperbolic η leaves invariant an isometrically embedded R (its axis Aη). Points on Aη are translated by `T (η):= min fdT (t; η(t)) j t 2 T g: • We will be interested in isometric actions of a fg group H on T . • The H-tree T is minimal if T contains no proper invariant H-subtrees. Lemma. If H is fg and T is a minimal H-tree, then T is either a point or the union of the axes of the hyperbolic elements of H. 2. Spaces of real trees •A (H)is the set of isometry classes of non-trivial minimal H- trees. • Gromov topology: limf(Tn; dn)g = (T; d) iff for all finite K ⊂ T , > 0, and finite P ⊂ H there are, for all large n, subsets Kn of Tn and bijections fn : Kn ! K such that: jd(ηfn(sn); fn(tn)) − dn(ηsn; tn)j < for all sn; tn 2 Kn and η 2 P . • Intuitively, larger and larger pieces of the limit tree with their restricted actions (approximately) appear in nearby trees. • We are interested in projectivized spaces of non-trivial H-trees, i.e. (T; d) ∼ (T; λd), λ > 0. PA(H) := A(H)=(0; 1) • Non-trivial h 2 Hom(H; F) gives an action of H on the Cayley tree for F and so determines Th 2 PA(H). • Since the Cayley tree is a free F-tree, if η 2 H fixes a point in Th, then h(η) = 1. • In particular, ker(Th) = ker(h). • T and T are isometric where i denotes conjugation by h iφ◦h φ φ 2 F. Theorem (Paulin, Culler-Morgan). Let fhig be a sequence in ( ). Then has a convergent subsequence in ( ). Hom H; F fThig PA H • Such limits have nice properties. • An H-tree T is super stable if the following property holds: if J ⊂ I are non-degenerate arcs with F ixT (I) non-trivial, then F ixT (J) = F ixT (I). • An H-tree T is very small if it is non-trivial (i.e. not a point), minimal, non-degenerate arc stabilizers are primitive abelian, and non-degenerate tripod stabilizers are trivial. Proposition. Suppose fhig is a sequence in Hom(H; F), no hi has cyclic image, and = lim . Then: T Thi 1. T is irreducible (i.e. not linear and no fixed end); 2. ( ) = ; and Ker T −!ker hi 3. T=ker(T ) is very small and super stable. In particular, H=ker(T ) is a limit group. Proof. We will show tripod stabilizers are trivial. The rest is similar. • Assume η stabilizes the endpoints of a tripod. Nearby have tripods with endpoints nearly stabilized by . • Thi η • So, η fixes the cone point and hi(η) = 1. 3. Shortening • Let H have a fixed finite generating set S. For h 2 Hom(H; F), khk := maxs2S jh(s)j. 0 0 • h ∼ h if h = iγ ◦ h ◦ a where γ 2 F and a 2 Aut(H). 0 • h is shortest if h = minh0∼h kh k. Theorem(Shortening) . Suppose H is fg, freely inde- composable, and not . If = lim where is a sequence Z T Thi fhig of shortest elements, then T is not faithful. 4. Proof of Theorem (K-M,R ) Theorem (K-M,S). Suppose H is fg and not free. Then there is a finite set S = fs : H Hsg of proper epimorphisms such that: • for all h 2 Hom(H; F); there exists a 2 Aut(H) such that h ◦ a factors through S. Further, if H is a not a limit group then we may always take a = IdH. Proof. • If H is not a limit group then there is a finite set fη1; : : : ; ηN g such that every h 2 Hom(H; F) kills some ηi. We may take S = fH=hhη1ii; : : : H=hhηN iig. • By Grushko, we may assume that H is freely indecomposable. We may also assume that H is non-abelian. • It is enough to show that there is S through which every short- est h 2 Hom(H; F) factors. Suppose not. • Let fη1;::: g enumerate the non-trivial elements of H. If Si := fH=hhη1ii; : : : ; H=hhηiiig then there is shortest hi that is injective on fη1; : : : ; ηig. A sub- sequence of fhig converges to a faithful tree contradicting The- orem(Shortening). Corollary. We may assume each Hs is a limit group. 5. Examples of factor sets • The sets that appear in the theorem are calledfactor sets. There aren't many explicit examples. • If H = H1 ∗ H2, then S(H) = S(H1) ∗ S(H2) := fs1 ∗ s2 j si 2 S(Hi)g. n n • S(Z ) = fs : Z ! Zg • If H is a closed, orientable, genus g surface group and if s : H ! Fg represents the standard retraction of the surface onto a rank g graph, then S(H) = fsg. (Zieschang, Stallings) • The non-orientable version is due to Grigorchuk and Kurchanov and one map does not suffice. Note: for n = 1, 2, or 3, nP is not a limit group. • We need to better understand real trees. We will use foliated 2-complexes to visualize real trees. Example (Triangles). }L }L Each colored band is decomposed into parallel line segments with a transverse Lebesgues measure..