Automaticity, almost convexity and falsification by fellow traveler properties of some finitely ...

178 Pages·2012·1.71 MB·English

Automaticity, almost convexity and
falsification by fellow traveler properties of
some finitely generated groups
Murray James Elder
Submitted in total fulfillment of the requirements
of the Degree of Doctor of Philosophy
June 2000. Revised September 2000
Department of Mathematics and Statistics
The University of Melbourne
Figure 1: Two Openings in Black over Wine, Mark Rothko 1958. The Tate
Gallery, London.
i
Abstract
We set out to examine the automaticity and almost convexity of an intrigu-
ing class of groups. Brady and Bridson provide examples from this class
with quadratic isoperimetric function that are not biautomatic. Thus show-
ing these examples are automatic would answer a long-standing question in
automatic group theory. Wise gives another example from this class which
is non-Hopfian and CAT(0). Determining the automaticity of this example
would answer one of two questions; are all CAT(0) groups automatic, and
are all automatic groups Hopfian? Determining its almost convexity would
give similar insight.
We start by trying to understand the geodesic structure of the Cayley
graphsoftheseexamples,foraparticularchoiceofgeneratingset. Thisleads
us to define the notion of a pattern in the Cayley graph, and we succeed in
characterising the set of all patterns for these groups. From this we can
prove they are almost convex for the chosen generating sets. This gives the
first example of a non-Hopfian almost convex group.
We also prove that the full language of geodesics is not regular, and
moreover there is no geodesic automatic language for these examples with
respect to the chosen generating sets.
Neumann and Shapiro define the falsification by fellow traveler prop-
erty and show that if a group enjoys this property then its full language
of geodesics is regular. Consequently the above examples do not enjoy this
property. Related to it is the loop falsification by fellow traveler property
which we introduce in this thesis. Figure 2 summarises some facts about
these properties. The two non-implications shown result from this thesis.
We ask whether all groups with a quadratic isoperimetric function enjoy
the loop falsification by fellow traveler property. If so we would have a
surprisingcharacterisationforthesegroups. AnexampleofStallingsappears
to provide some clues to this question.
Wealsoexaminethequestionofhigherdimensionalfinitenessandhigher
dimensionalisoperimetricfunctionsforgroupsenjoyingthesegeometricprop-
erties. We prove that if a group enjoys the falsification by fellow traveler
property then it is of type F . We ask whether the larger class of almost
3
convex groups are of type F . Stallings’ group would be a potential coun-
3
terexample to this, since it is finitely presented and not of type F . We
3
prove that for two independently arising generating sets, Stallings’ group is
ii
full language
of geodesics
is regular
?
finitely
presented
falsification by almost
fellow traveller convex
at most
property
exponential
isoperimetric
function
finitely
presented
loop asynchronous loop
falsification by falsification by
at most
fellow traveller fellow traveller
quadratic
property ? property ? isoperimetric
function
Figure 2: Implication diagram
not almost convex, suggesting it is not almost convex for anygenerating set.
iii
This is to certify that
(i) the thesis comprises only my original work,
(ii) due acknowledgment has been made in the text to all other material
used,
(iii) the thesis is less than 100,000 words in length.
iv
Acknowledgments
MythankstoWalterNeumannforhisencouragement, ideasandadvice, and
thanksto Mike Shapiroandto AndrewRechnitzer fortheir helpthroughout
my PhD. Thanks to Dean Chequer, Mum and Dad, Alisoun, Bill, Kathy,
Esther, Lisa, Megan, Jessica, Suzanne Buchta, Lois Bedson, Janie Burrows,
Bell Foozwell, Paul Gregg, Averil Newman, Amanda Johnson and Kerry
Williams for their love, patience and support, and thanks to Noel Brady,
Martin Bridson and Sarah Rees for their invaluable suggestions.
v
Contents
1 Introduction and Definitions 1
1.1 Some Open(ing) Questions . . . . . . . . . . . . . . . . . . . 1
1.2 Geometric Group Theory . . . . . . . . . . . . . . . . . . . . 4
1.3 Almost convex groups . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Isoperimetric functions . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Regular languages . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Fellow traveler properties . . . . . . . . . . . . . . . . . . . . 15
1.7 Automatic groups . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 Changing weighted generating sets preserves automaticity . . 32
1.9 Some other miscellaneous geometric group theory . . . . . . . 35
1.9.1 Eilenberg MacLane spaces . . . . . . . . . . . . . . . . 35
1.9.2 CAT(0) groups . . . . . . . . . . . . . . . . . . . . . . 36
1.9.3 HNN extensions . . . . . . . . . . . . . . . . . . . . . 36
1.9.4 Hopficity . . . . . . . . . . . . . . . . . . . . . . . . . 38
2 Automaticity and almost convexity for an example of Brady
and Bridson 39
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Asynchronous automaticity . . . . . . . . . . . . . . . . . . . 41
2.3 Geodesic structure and Patterns . . . . . . . . . . . . . . . . 46
2.3.1 Geodesics and “Pre-sequences” . . . . . . . . . . . . . 47
2.3.2 Sequences and Patterns . . . . . . . . . . . . . . . . . 50
2.3.3 An imaginary finite state automaton . . . . . . . . . . 52
2.3.4 Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.5 Moves as rewriting rules. . . . . . . . . . . . . . . . . 60
2.3.6 Proof of the Conjecture . . . . . . . . . . . . . . . . . 63
2.4 Almost convexity . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5 Geodesic automatic languages . . . . . . . . . . . . . . . . . . 80
vi
3 The Wise Group 85
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 Asynchronous automaticity . . . . . . . . . . . . . . . . . . . 86
3.3 Geodesic structure and Patterns . . . . . . . . . . . . . . . . 90
3.4 Geodesic automatic structures. . . . . . . . . . . . . . . . . . 103
3.5 Proof of the Conjecture . . . . . . . . . . . . . . . . . . . . . 107
3.6 Almost convexity . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4 Finiteness properties, isoperimetric functions, the falsifica-
tion by fellow traveler property and almost convexity 135
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 Finiteness for asynchronously automatic groups . . . . . . . . 136
4.3 Groups with the falsification by fellow traveler property are
of type F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3
4.4 Almost convexity and Stallings’ example . . . . . . . . . . . . 148
4.4.1 Baumslag et al’s presentation . . . . . . . . . . . . . . 148
4.4.2 Bestvina and Brady’s presentation . . . . . . . . . . . 152
4.4.3 Bridson’s presentation . . . . . . . . . . . . . . . . . . 154
4.5 Theloopfalsificationbyfellowtravelerpropertyandquadratic
isoperimetric functions . . . . . . . . . . . . . . . . . . . . . . 155
vii
List of Figures
1 Two Openings in Black over Wine, Mark Rothko 1958. The
Tate Gallery, London. . . . . . . . . . . . . . . . . . . . . . . i
2 Implication diagram . . . . . . . . . . . . . . . . . . . . . . . iii
1.1 Γ (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
{a,b} 2
1.2 Γ (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
X 2
1.3 Making the graph ΓX(cid:48)(G) . . . . . . . . . . . . . . . . . . . . 6
1.4 γ of length k+m . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Length at most C(k)2 . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Growth of the metric ball in a non-almost convex group . . . 10
1.7 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.8 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.9 The Pumping Lemma . . . . . . . . . . . . . . . . . . . . . . 15
1.10 The falsification by fellow traveler property . . . . . . . . . . 16
1.11 Geodesics asynchronously k-fellow traveling . . . . . . . . . . 17
1.12 Case 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.13 Case 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.14 Case 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.15 Case 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.16 Case 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.17 Case 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.18 Case 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.19 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.20 Showing L has the fellow traveler property . . . . . . . . . . 34
Y
1.21 Padding the words so that they fellow travel . . . . . . . . . . 35
1.22 Cyclic subgroups of (cid:0) 2 . . . . . . . . . . . . . . . . . . . . . . 37
2.1 π (S1)= (cid:0) = (cid:104)a(cid:105), B = (cid:104)a,t |t−1a2t= a3(cid:105) . . . . . . . . . . 40
1 2,3
2.2 π (T) = (cid:0) 2 = (cid:104)a,b | ab = ba(cid:105). We choose cyclic subgroups
1
(cid:104)a(cid:105), (cid:104)ab(cid:105) and (cid:104)ab−1(cid:105). . . . . . . . . . . . . . . . . . . . . . . . 40
viii
2.3 A presentation 2-complex for (cid:104)a,b,s,t | ab = ba,s−1as =
ab,t−1at =ab−1(cid:105). . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 A 2-combing of (cid:0) 2 . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 r = s−1,x =ab . . . . . . . . . . . . . . . . . . . . . . . . . 43
n
2.6 r = s,x= b . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
n
2.7 r = s,x= ab−1 . . . . . . . . . . . . . . . . . . . . . . . . . 44
n
2.8 A (cid:0) 2 plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9 An s-strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.10 Coset representatives for a normal form language . . . . . . . 47
2.11 w = c2b,r = s−1 . . . . . . . . . . . . . . . . . . . . . . . . 48
0 1
2.12 (All) geodesics to the first strip. . . . . . . . . . . . . . . . . . 48
2.13 The next plane for w =c2s−1d−2bs−1... . . . . . . . . . . . 49
2.14 Determining the next pre-sequence for w. . . . . . . . . . . . 50
2.15 Exiting the second plane by a different strip. . . . . . . . . . 51
2.16 “Initial patterns” (−1)(0)(1) . . . . . . . . . . . . . . . . . . 54
2.17 “Parallel moves” . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.18 Move 2 gives (−1)(0)(10)(1) . . . . . . . . . . . . . . . . . . . 56
2.19 Another move 2 gives (−1)(0)(10)(1110)(1) . . . . . . . . . . 57
2.20 Another move 2 gives (−1)(0)(10)(1110)(170)(1) . . . . . . . 58
2.21 A potentially “bad” pattern . . . . . . . . . . . . . . . . . . . 59
2.22 Move 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.23 Move 2: c/d →a . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.24 Move 3: a→c/d . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.25 Move 4: c/d →d/c . . . . . . . . . . . . . . . . . . . . . . . . 63
2.26 γ = r r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1 2
2.27 γ = rx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.28 Last strip is s,t . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.29 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
1
2.30 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2
2.31 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3
2.32 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4
2.33 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5
2.34 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6
2.35 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7
2.36 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8
2.37 Last strip is s−1,t−1 . . . . . . . . . . . . . . . . . . . . . . . 72
2.38 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1
2.39 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2
2.40 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3
2.41 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4
ix