Thompson s Group F (p + 1) is not Minimally Almost Convex

Transcription

1 Thompso s Group F (p + ) is ot Miimally Almost Covex Claire Wladis Thompso s Group F (p + ). A Descriptio of F (p + ) Thompso s group F (p + ) ca be defied as the group of piecewiseliear orietatio-preservig homeomorphisms of the closed uit iterval with breakpoits i Z[ p+ ] ad slopes which are powers of p + i each liear piece Figure : Oe elemet of F (p + )

2 . Represetatio of F (p + ) by tree-pair diagrams p - - -,,p - T_ T+ p+,,p - p p p+,,p - p,,p - p Figure : The homeomorphism of the closed uit iterval ad the tree-pair diagram represetig x i F (p + ).. Defiitios A caret i a (p + )-ary tree has (p + ) vertices. Oe of these vertices is called the paret; the paret is coected by ( + ) directed edges from itself to each of the other vertices i the caret, which we call the childre of the paret vertex. Two childre of the same paret are called sibligs. A graph formed by joiig ay umber of (p + )-ary carets is referred to as a (p + )-ary tree. The topmost caret is referred to as the root caret (or just the root) ad it s paret is called the root ode. If a vertex i the tree has o childre, it is referred to as a leaf; if it does have childre, it is referred to as a ode. A elemet of F (p + ) ca be represeted by a (p + )-ary tree pair diagram. A (p + )-ary tree pair diagram is a pair of (p + )-ary trees cotaiig the same umber of odes. The first tree i the pair is the egative tree ad the secod tree i the pair is the positive tree. This pair of trees is deoted (T, T + ). The odes of each tree are umbered, ad the th ode of the egative tree is paired with the th ode of the positive tree.

3 .. Leaf Orderig i a Tree-pair Diagram We ca umber the leaves of each of the trees i a tree-pair diagram of a elemet of F (p + ) by thikig of each leaf as a subiterval of the closed uit iterval; we umber the leaves of the tree from left to right with respect to their positio as subitervals of the closed uit iterval p+,,5p- T_ T p +,..., 7 p 3p+,,4p p+,,3p 6 5 p,... 6 p,..., 7 p p 5,5p 5p+,,5p+3 3,p 4,,p p+,,p+4 3,4p 4p+,,4p+3,3p 3p+,,3p+3,p p+,,p+3 3,,p p+,,p+3 Figure 3: x x 5 3x 4 x 3 i F (p + ) with all carets ad leaves umbered

4 .3 Presetatios of F (p + ) F (p + ) stadard ifiite presetatio: F (p + ) = {x, x, x j x i = x i x j+p for i < j},,p-,,p- p p+,,p p p+,,p x xi, i..,p p+,,p i,,p p+,,p+i,,p,,p-,,p-,,p- xp p p+, p,..., 3 p,,p- p p+,,3p p+,p xi p p+,,(+)p,,p- - xp (-)p,,(+)p (-)p+i,,p p+,,p+i p + p+,,,p- (+)p,,(+)p +,(+)p p p+,,(+)p Figure 4: The geerators {x, x, x, } for the stadard ifiite presetatio of F (p + ) (i =,, p ad N) F (p + ) stadard fiite presetatio: F (p + ) = {x, x,, x p [x x i, x k ]} where k = p +,, p + ad i =,, p.,,p-,,p-,,p- x p p+,,p p+,,p xi p p+,,p,,p xp p p+, p,..., 3 p,,p i,,p p+,,p+i,,p- p+,,3p p p+,p Figure 5: The geerators {x, x,..., x p, x p } for the stadard fiite presetatio of F (p + ) (i =,, p )

5 .4 Fordham s carat types, ode/caret orderig i treepair diagrams, ad his method for computig word legth i F (p + ). Table : Weight of Types of Caret Pairs i the Tree-Pair Diagram of Elemets i F (p + ) (j i < j, i < j i ) (, ) L R R R R j M i M i j L R 3 R R 3 R j 3 3 M i M i j R j 3 M i 3 4 M i j We will use the otatio w( i ) to deote the weight, give by Fordham s table, of the pair of carets i the tree-pair diagram umbered i. We ote that carets of type L are ot listed o the table; sice there is oly oe caret of this type i both the positive ad egative trees, the oly pairig possible is (L,L ) ad w(l, L ) =. Theorem.. (Fordham). Give a elemet w i F (p + ) described by the reduced tree pair diagram (T, T + ), the word legth w of the elemet with respect to the geeratig set {x, x,..., x p, x p } is the sum of the weights of each of the pairs of carets i the tree pair diagram.

6 .4. Multiplyig tree-pair diagrams Simplifyig tree-pair diagrams: We say that a caret is exposed if it has o child carets (i.e. all its childre are leaves). If there exist exposed carets with exactly the same umber o both the positive ad egative trees the these two carets ca be removed from their respective trees without chagig the elemet that the tree-pair diagram represets. This removal of uecessary carets is aalogous to simplifyig a word. Multiplicatio is o the right.,... p,... xp,......, 3 p,... p,... p x..., p,... p p,..., 3 p p,... p,... p,... xp,......, 3 p =,......, 3 p x..., 3 p,... p,..., 3 p p,... p p,... p p,... p,... xxp..., 3 p p,...,... p,..., 3 p p,... p Figure 6: Multiplicatio of tree-pair diagrams for the product x x p i F (p + ) (to complete multiplicatio, oe caret must be added to the leaf umbered p i the tree-pair diagram for x )

7 Almost Covexity We let B deote the ball of radius i the Cayley graph Γ of a group G with the fiite geeratig set X. The covexity fuctio c() of the group is defied to be c() = max{d B (g, h) g, h B ad d Γ (g, h) = }. Defiitio (almost covex).. (Cao). A group G is almost covex with respect to the fiite geeratig set X if its covexity fuctio c() with respect to the fiite geeratig set X is bouded by a costat C, i.e. c() C for all. Defiitio (miimally almost covex).. A group G is miimally almost covex with respect to the fiite geeratig set X if its covexity fuctio c() with respect to the fiite geeratig set X is bouded by the costat for sufficietly large, i.e. if there exists a costat N such that for all > N, c(). Sice there is always a path from g to h i B through the idetity which has legth i B bouded by (g h, for example), we will always have c(). Belk ad Brow have already proved that F () (usually deoted F ) is ot miimally almost covex; the proof that F (p + ) is ot miimally almost covex is a geeralizatio of that proof.

8 . Thompso s group F (p+) is ot miimally almost covex L(g) is used to deote the legth of a elemet g i the group with respect to the give geeratig set. w( i ) is used to deote the weight of the pair of carets which have idex umber i i the tree-pair diagram of the elemet. Theorem.3. For all there exist l, r, F (p + ) such that with respect to the stadard fiite geeratig set {x, x,..., x p, x p }:. d(l, r) =. L(l) = L(r) = + 3. Ay path p from l to r which remais i B + is such that L(p) We choose r = x p x (+) x p ad l = rx T_ (+)p+,,(+)p-,..., p - + (-)p+,,(+)p ( +3)p+,,(+4)p + (+)p+,,(+)p- T+ +,,p ( +)p,,(+3)p r p+,,(+)p + 3 ( +)p,,(+3)p- ( + 3 ) p ( +3)p+,,(+4)p p p+,,p - - T_ (-)+,,p- + +,..., p - (-)+,,(-)p p,,(+)p- + 3 T+ + (+)p,,(+)p- (+)p+,,(+)p- + (+3)p+,,(+4)p + 3 p+,,(+)p ( +)p,,(+3)p-,,p ( +)p,,(+3)p ( + 3 ) p ( +3)p+,,(+4)p l p p+,,p Figure 7: r (top) ad l (bottom) i F (p + )

9 .. Vertexes o the Path from l to r with distace + 3 We let p represet the path from l to r which does ot leave B +. Lemma.4. (Belk ad Bux) With respect to the stadard fiite geeratig set for F (p + ), if there are two vertices h r ad h l o the path p from l to r such that d(h l, h r ) + 3, the L(p) Defiitio (right foot).5. Let s be the first right caret i the egative tree.. If s has a oempty left subtree, the we cosider this subtree: withi this ew subtree, the right foot is the rightmost child of the last right caret i the subtree, which will be a leaf (if it were ot a leaf, there would be aother right caret i the tree).. If s has a empty left subtree, the the right foot is the leftmost leaf of s i the egative tree. 3. If the egative tree has o carets of type R, the the right foot is the rightmost leaf of the root caret i the egative tree. Defiitio (critical leaf).6. The critical leaf is the same thig as the right foot; the oly differece is that the right foot is a leaf o the egative tree ad the critical leaf is a leaf o the positive tree. 3 4 p p+,,p- 6,5p 5p+,,5p+3,4p 4p+,,4p+3 6 p p 5,6p 6p+,,6p+3 7 7p+,,8p 4 6 p +,..., 7 p 3 3p+,,4p p+,,3p 3,p 4,,p p+,,p+4 5 4p+,,5p- 6 5 p,... 6 p,..., 7 p + + ( +)p+,,(+3)p,,p,3p 3p+,,3p+3 p+,,p p+,,p+3 Figure 8: Right foot/critical leaf idicated by arrow i several (p + )-ary trees

10 For the elemet l, the right foot is to the left of the critical leaf (idex of right foot is p ad idex of critical leaf is ( + )p), ad for the elemet r, the right foot is to the right of the critical leaf (idex of right foot is ( + 3)p ad idex of critical leaf is ( + )p). O the path p from l to r: The right foot crosses the critical leaf oce we have the idex of the the right foot greater tha or equal to the idex of the critical leaf. The right foot hits the critical leaf whe they both have the same idex umber... Fidig the vertex h r Lemma.7. Ay path p from l to r which remais i B + must pass through the vertex rx p x. p+,,(+)p- + T_,..., p - + T+ (+)p,,(+)p- ( +)p (+)p+,,(+)p- ( +)p,,p p+,,(+)p p p+,,p Figure 9: h r i F (p + ) If we let h r = rx p x, h r is the last vertex alog p o which the right foot hits the critical leaf.

11 ..3 Fidig the vertex h l We let h l be the first vertex alog p o which the right foot hits the critical leaf. Lemma.8. For ay vertex o the path p from l to r o which the right foot has ot yet crossed the critical leaf, the last 3 carets i the positive ad egative trees of that vertex will remai the same as they were i l (see figure ). These are the carets which have a idex umber higher tha the caret cotaiig the critical leaf. B a,..., a p C C p- c,..., c p + p- p- c,,cp+ D T_ T+ E f f,..., f p c,,cp b b,,bp c d,,..., d, p e,..., e p d s,,..., d s, p Figure : A vertex o the path p from l to r at which the right foot has ot yet crossed the critical leaf (after carets have bee added as eeded so that it ca be multiplied by each of the geerators) Lemma.9. The first vertex o the path p from l to r at which the right foot crosses the critical leaf is also the first vertex at which the right foot hits the critical leaf (i.e. the right foot caot cross the critical leaf for the first time o the path without hittig the critical leaf).

12 ..4 Left-Sided Elemets Defiitio (left-sided).. A elemet f of F (p + ) is left-sided if:. The idex umber of the right foot is the same as the idex umber of the critical leaf (i.e. t = s), ad. All right carets i each tree are of type R. w T_ T+ y w,..., w p y,..., y p x,..., x p z,..., z p Figure : A left-sided elemet i F (p + ) Remark.. h r is left-sided. Defiitio.. We defie y = x x px. Remark.3. Left-sided elemets commute with y. Lemma.4. For ay left-sided elemet f i F (p + ), L(fy) = L(f) + 3. w w,..., w p T_ y y,..., y p T+ x,..., x p z,..., z p Figure : yf = fy for left-sided elemet f i F (p + )

13 Lemma.5. There exists a left-sided elemet h l o the path p from l to r such that h l = yh l. B a a,..., a p - T_ T+ E f d, c,,cp f,..., f p d,,..., d, p e,..., e p d s,,..., d s, p Figure 3: Simplified versio of a vertex o the path p from l to r at which the right foot has ot yet crossed the critical leaf a T_ T+ f a,..., a p f,..., f p d. e Figure 4: h l i F (p + )

14 Lemma.6. The tree-pair diagram of h r h l is already reduced. Figure 5: h r h l i F (p + ) Defiitio (width of a group elemet).7. Let m deote the umber of carets i either the positive or egative tree of the tree pair diagram of f i F (p + ). The width of f, deoted by w(f), is equal to m. Lemma.8. For ay left-sided elemet f of F (p + ), L(f) w(f). Remark.9. h r h l is left-sided. Corollary.. L(h r h l ) Theorem.. F (p + ) is ot miimally almost covex.