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Geometrical Finiteness
for Hyperbolic
Groups
B rian
H ayw a rd
B ow d itch
Thesis submitted for the degree or Doctor or Philosophy at the
U niversity or Warwick.
M athem atics In stitu te ,
U n iv e rsity or W arw ick,
Coventry, C V4 7AL,
United Kingdom.
D e cla ratio n
Summary
A c k n ow le d ge m e n ts
G e o m e t r ic a l F i n i t e n e s s f o r H y p e r b o l i c G r o u p s Brian H. Bowditch
University o f W arwick, Coventry, C V 4 7AL, U.K .
A b s t r a c t .
Let T be a g ro u p acting properly discon tin u ous!/ on hyperbolic space H " . T he aim o f this work is to clarify the meaning o f “geom etrical finiteness” fo r such groups. In dim ension 3, with T acting freely, the principal definition is that T should possess a “ finite-sided fundamental d om a in ". Various definitions in this dimension have been worked out. Marden [Mar) shows that it is equivalent to the statement that the quotient manifold, including its ideal points, can be decom posed into a com pact part and standard neighbourhoods o f its cusps ( “ cusp cylinders" and “ cusp to ri"). Thurston [T h I] introduced tw o new definitions: that the th ick part o f the convex core should be com pact, o r that an « -neighbourhood o f the convex core should have finite volume. Finally, Beardon and Maskit [BeaM] say that T is geom etrically finite if and only if its limit s e t consists o f (w h at we call here) conical limit points and bounded parabolic fixed points.
W e shall investigate here how these definitions generalise to arbitrary discrete actions in n dimensions. T h is matter has also been considered by A panasov [A p l,A p 2 ], and to som e extent by W eilenberg |We|.
Our central definition (G F l) will be essentially that o f Marden, with appropriate definitions o f cusp regions. The B eardon and M askit description (as they p oint o u t in their paper) generalises unchanged (G F 2 ). T h e use o f finite-sided fundamental domains runs into problem s when n > 4. T he natural generalisation seem s to be in term s o f what we call “ convex cell com plexes" (G F 3 ). For the first o f T hurston’ s definitions, w e need to clarify w hat we mean by the “ thick p a rt" o f an orbifold. T h e definition chosen here (G F 4 ) does n o t seem particularly natural, but it proves useful in discussing the final definition (G F 5 ). For this, we im pose the additional condition that T be finitely generated. W e suspect that this is unnecessary, and show it t o be unnecessary for m anifolds, finite-volume orbifolds, o r when n < 3. In the course o f this discussion, w e give a p roof o f th e existence o f an embedded ball in a hyperbolic n-orb ifold, o f uniform radius, depending o n n , but not on th e particular orbifold.
Finally, in C h .4 , we discuss the existence o f finite-sided fundam ental domains. W e give (in principle) a com plete description o f when a Dirichlet region is finite-sided, and show that in certain special cases, all c o n v e x fundamental domains are finite-sided. W e give an exam ple (due t o Apanasov) o f a geometrically finite (henceforth abbreviated to G F ) manifold w ith no finite sided Dirichlet domain.
A c k n o w le d g e m e n t s .
This work w as originally an offshoot from the Warwick M .Sc. dissertation o f Dick C anary and Paul G reen , which has been published, in augmented form , as [CanEG ]. Their paper focuses o n other aspects o f Thurston’ s notes [T h l], t u t the start they m ade on geom etrical finiteness was very helpful to me. My greatest debt is t o D avid Epstein, for introducing me to the subject, and for his many suggestions and com m ents. I w ould also like to thank the S.E .R .C . for their financial support.
O. I n t r o d u c t io n .
0 . 1 . H y p e r b o lic S p a c e .
W e begin w ith a general discussion o f hyperbolic geom etry in order t o induce our term inology and notation. More deta ils may be found in [Bea, C hapter 7|.
isom etries o f X .
W e can represent H n conform ally mm -he o p en unit ball in R " with infinitessimal metric dhvp = ^ r d , HC, where r is the euclidean distance from the centre. T h is is the P o in ca r i model. T h e closed unit ball gives a canonical com pactification o f H n, which we d en ote by H £ . W e write H ? for the (n - l)-sphere o f ideal p oin ts, so that H £ = H n U H " . A n y isometry 7 G I s o m H " can b e extended to a ct conform ally on H " .
Another conform al representation o f H " is as the upper half-space in R n ; that is, I t " = { * e I t " | x n > 0 } , where x „ is the last coordinate o f x . T h e m etric is given infinitessim ally b y dhpp = ^ -d ,ue. Writing 3 R " = { i E R " | x „ = 0 } , we may identify H " as 5 R £ U {0 0} , where the ideal point 00 compactifies R " U 3 R " into a balL Note that if 7 € I s o m H " fixes 00, then it acts as a euclidean similarity on 3 R " .
A third m odel for hyperbolic space we shall use is the K lein model. T h is consists o f the open unit ball w ith a (non-conform al) Riemannian metric, such th a t all hyperbolic geodesics correspond to euclidean line segm ents (see [Bea, C hapter 7 j).
W e m ay classify non-trivial isometries H " in to three types, nam ely elliptic, parabolic and hyperbolic as follow s.
L et 7 be an isometry o f H " . W e write 6x7 fo r the set o f fixed points o f 7 in H ".. Brouwer's fixed point theorem tells us that fix 7 must be non-em pty.
Suppose that there is som e point x in 6x7 0 H " . We m ay take x to b e the centre o f the ball in the P oincari model. Then, 7 a cts as a euclidean rotation on the ball, and we see that 6x7 is a (possibly 0-dim ensional) plane in H ".. W e call this case elliptic.
If 7 is not elliptic, then 6x7 is a subset o f H " . Suppose that 6x7 consists o f ju st a single point in H " . W e m ay take this poin t to be 00 in the upper h alf space m odel, R " . Now, since 7 has no fixed point in d R " , it m ust act as a euclidean isom etry o f d R !J . M oreover, it must preserve setwise each horosphere ab out 00. W e ca ll this case parabolic.
Suppose that 7 fixes precisely two points, x a n d y, in H7. Let l b e the geodesic joining x to y . In this case, 7 acts as a translation on I, and (in general) has a rotational com ponent in the orthogonal direction. W e c a ll this case lozodrom ie, and we call l the loxodrom ic axis.
Finally, note that if 7 has three (or more) fixed points in H " , then these must determine a fixed point in H " , so we are back in the elliptic case.
0 . 2 . G r o u p « o f I s o m e tr ie s .
L et T be a subgroup o f I s o m H " . It is an elem entary result that T is a discrete subgroup if and on ly if it a cts properly discontinuously on I I " , that is t o say, each com pact subset o f H " meets only finitely many im ages o f itself under I\
In such a discrete group, the finite-order elem ents are precisely the elliptic isometries. Thus, T acts freely if and only if it is torsion-free. If T acts freely, w e may form the quotient m anifold M —
H "/r
which in herits a c om plete hyperbolic structure.M ore generally, if T has torsion, the quotient M = H n/ r is a c om plete hyperbolic “ orbifold*, as defined b y T hurston (T h l, chapter 13). T h a t is to say, there is a closed cell com plex £ in M , such that A f\ E is an (incom plete) hyperbolic manifold. T he set E can b e defined as the projection o f the set o f all fixed points o f e lliptic elements o f T, Le., E = U -,er(® x 'Tn H " ) / I \ A neighbourhood o f a poin t o f E m ay or m ay not be topologically singular, but it will always be geom etrica lly singular. In an orientable 2-orbifold, for example, E consists o f a discrete set o f cone singularities, w hich m ay be thought o f as points o f concentrated positive curvature. W e shall call E the singular set o f A /.
Let T C I s o m H " be discrete. The action o f T m a y be e xtended to H £ , and we m ay define the limit set A C H " as the set o f accum ulation points o f som e T -orbit, i.e.,
A = ( y € H " | there exist 7* € T and x e H " with 7„ x —• y } .
It turns out that this definition is independent o f our choice o f x . M oreover, A is a minim al closed V- invariant set, and T acts properly discontinuously on its c om plem ent O in H " . T h e set f l = H "\ A is called th e discontinuity domain. (It is possible for i l t o b e em pty.) W e may form the quotient orbifold A // — (l/T
o f i) . Since T acta conform ally on H " , we tee that M , inherit* a (singular) conform al structure from fl. In fa ct, T a cts properly discontinuously on H " U Q, so w e m ay write
m c = ( H B u
n)/r
= M u M , .N ote th a t when n = 3, A»',- is a Riemann surface (in general not connected). T h is fact gives rise to a rich ana lytical theory in this dir tension.
O ne direction o f research in discrete h yperb olic groups, is study to the relationship o f various types o f ‘ finiteness” — group theoretic, topological and geom etric. T h e simplest grou p theoretic restriction is to dem and th a t T be finitely generated, and ask w hat this tells us ab out the top olog y and geom etry o f M .
T h e first result is pure algebraic.
S e l b e r g L e m m a |Selj. L e t k be a field o f characteristic 0. Then, any finitely-generated subgroup o f G L n(fc) is virtually torsion-free, (i.e . contains a torsion-free subgroup o f finite index).
F or a sim pler proof, see [Cas].
Since Isom H " can be represented as a subgroup o f G L n+ i(R .), the Selberg Lem m a can b e applied to finitely-generated subgroups o f Isom H " . Geom etrically, this tells us that w e can restrict attention to the case where T a cts freely on H n . G iven this, we m ay as well assume also that I* preserves orientation. This latter restriction is solely t o sim plify the exposition. T hus, for the rest o f the introduction, unless otherwise stated, w e shall be taking T to b e a finitely-generated, discrete, torsion-free group o f orientation preserving isom etries o f H n .
B eyon d th e Selberg Lemm a, little seems t o b e know n in general. T h e main thrust o f research is in dim ension 3, and we shall give a summary o f 3-dim ensional results in Section 0.3. First, w e describe how the 2-dim ensional case is trivial from the point o f v iew o f finiteness.
Let M b e a com plete, orientable, hyperbolic surface w ith finitely-generated fundam ental group. Then, it turns o u t that M consists o f a com pact surface with boundary, together w ith a finite num ber o f “ cusps* and ‘ funnels*. A cusp is (isom etric to) a horoball in H 3, quotiented ou t b y a cyclic parabolic group (F IG 0.1). A fu n n e l consists o f a hyberbolic half-space quotiented o u t by a loxodrom ic element (F IG 0.2). W e see that M i is a disjoint union o f finitely m any circles, w hich serve t o com pactify th e funnels in M e = M U M i. T h u s the top ological ends o f M e correspond precisely t o the cusps (FIG 0.3). W e see that, in any meaningful sense, the geom etry o f M is on ly finitely com plicated. T h is is ab out the strongest assertion o f finiteness one cou ld make.
0 . 3 . S o m e 3 -d im e n s io n a l fin ite n e s s r e su lts.
In this section, we shall give a summary o f som e finiteness results in 3 dim ensions. It is not meant to reflect the historical developm ent o f the subject.
L et T b e a discrete, torsion-free, orientation-preserving subgroup o f Isom H " . M uch o f the technical com p lica tion o f the subject arrises from having to deal w ith parabolic subgroups o f I*. S uppose that 7 € T is parab olic w ith fixed poin t p. Let Tp b e the stabiliser o f p in I*. In a discrete group, a parabolic and a loxodrom ic can n ot share a c om m on fixed point. T hus, Tp consists entirely o f parabolics. W e c a ll p a parabolic
fixed p oint, w hich we a bbreviate to p.f.p. W e can let p be the point 00 in the upper half-space m odel. Now, r „ acts freely as % group o f isometries on d R * 3 E 3. T h is dimension is special in that such a group must a ct by translation. W e see that Tp is isomorphic t o either Z or Z ® Z . Taking l i to be any horoball ab out p, we m ay form the quotient B/Tp. If 3 Z , then BB/Tp is a bi-infinite euclidean cylinder, and we call
B /Tp a Z -cu s p (F IG 0.4). (N ote that a Z -cu sp is n ot quite the same as a “ cu sp cylinder* o r ‘ standard cusp
region* w hich will be described later in this section. See Section 2 for details.) If Tp 2! Z © Z then BB/Tp is a euclidean torus, and w e call B/Tp a Z © Z -cu sp (F IG 0.5). W e may define such cusps t o correspond to each orb it o f parabolic fixed point*. In general, on e w ould exp ect these cusps to project t o a collection of im mersed subm anifolds in M . However the Margulis Lem m a (see Sections 1.2, and 2 .(G F 4 )) tells us that (in dim ension 3 ), by taking our horoballs small enough, we can arrange that the cusps be d isjoin t and embedded in M . W e shall write c u sp (A f) for the disjoint union o f a ll the cusps.
T h e construction o f this set o f disjoint cusps is valid for infinitely-generated groups. From now on, how ever, w e shall insist that T be finitely-generated. W e first use a purely topological result.
T h e o r e m (S c ott [Sc)). L e t M be a 3-manifold with finitely-generated fundamental group. Then, there i t a
c om pact submanifold M T o f M , such that the inclution M-p A f induces an isom orphism o f fundamental
groups.
W e c a ll Mt > topological core tor M. W ith Af = H " / r , we deduce im mediately that T is finitely presented.
In our case, M = H n/ r is an irreducible 3-m anifold, that is each em bedded 2-sphere in A f bounds a 3-ball. Because o f this, we can arrange that d M r contains n o 2-spheres, and then the inclusion o f Mt in to M is a h o m o to p y equivalence. M oreover, there is a bijective correspondence between the boundary com ponents o f Mt and the topological ends o f M . W e deduce that M has only finitely m any ends. In particular, it contain s o n ly finitely many Z © Z-cusps.
In fa c t (provided that T is not cyclic loxodrom ic), the Z © Z -cusps correspond precisely to the toroidal c om p onen ts o f dM p - T* e remaining ends correspond to com ponents o f genus at least 2. T he aim now is to understand something o f the geom etry o f these remaining ends, which we shall call "non-cuspidal ends*.
N ow , a Z -cu sp is topologically ju st a product. Thus, w e can assume that each Z -cu sp lies entirely within som e n on-cuspidal end. T he effect o f removing the Z -cu sp s w ould (in general) be to subdivide each each such end in to smaller pieces, on which w e may see q ualitatively different behaviour. It is therefore necessary t o ta ke a ccoun t o f these Z-cusps before going on to consider the geometry. W e can d o this by applying a relative version o f S cott’ s theorem to M ' = M \cusp(A f).
T h e o r e m [ M e ] . Let N be a 3-manifold with boundary, whose fundamental group is finitely generated. L et S
be a com pact submanifold o f d N . Then, we can find a topological core, Np$ fo r N such that N y f i d N = S .
B y using this result, together with an Euler characteristic argument, one m ay deduce (FM| that there are on ly a finite number o f Z-cusps — a resu’ t due originally to Sullivan [Sull|. We may now take a core M'T o f M ' w hich meets each Z © Z -cu sp in the bounding torus, and each Z -cu sp in a com p a ct annular core o f its bou n d a ry cylinder. Again, we m ay take the inclusion t o be a (relative) hom otopy equivalence, so that the to p o lo g ic a l ends o f M ' correspond to the frontier com ponents o f M f in M '. We now look for geometric in form a tion ab out the non-cuspidal ends o f M ' (i.e. ends other that Z © Z -cusps).
W e have already said that, for n = 3, M / — f l / T is a Riem ann surface. A fundam ental result about M ; is th e follow ing.
A b l f o r s ’ F i n i t e n e s s T h e o r e m (A h l.S u ll). L et 1' be a finitely-generated discrete subgroup o f Iso m H 3.
T hen M i = i l / r is a Riem ann surface o f *finite type*. That is to say, M i is conform ally equivalent to a com /act surface with finitely many punctures.
(F o r a p r o o f using deform ation theory, see [Sul4|.)
M oreover one m ay show that the punctures o f A f/ arise only from parabolic elem ents o f T; that is, a sm all lo o p around a puncture represents a conjugacy class o f p arabolics in T.
W e w ant to give Ahlfors’ Finiteness Theorem a m ore geom etric interpretation. We can d o this by using the c o n v e x h ull o f the limit set — a generalisation o f the Nielsen convex region in dim ension 2. Let Y be the sm allest c onvex set in H3 whose closure, Y c , in contains the lim it set A. T hen, Y a m eets H ] precisely in A. Since the construction is equivariant, we may form the quotient ^ = Y/V Q M , which w e call the convex
core o f M . T h e nearest point retraction o f H3 on to Y extends continuously t o all o f H * , and therefore gives rise to a m a p from M e to ^ (see for exam ple (T h lj). W e shall denote by q, the restriction o f this m ap to
M ,. N ote th a t q (M ,) = 3 ? .
Finiteness Theorem to say that d f ' should have finite 2-dim ensional area. (In fact the discussion applies equally well if T has torsion, and then d ? becom es a finite-area orbifold.)
T h e parabolic cusps o f the h yperb olic surface d P are essentially the connected com ponents o f d f ' n c u sp (A f). In fact the cusps o f dH must lie inside Z-cusps o f A f. T h e remainder o f d f , namely d ¥ n A /', is com pact. T hus, each com ponent o f d V corresponds to an end o f A f'. Such an end is topologically a product, being foliated by c om ponents o f d N rf t ) for r > 0, where ATr ( ^ ) is a uniform r-neighb ourhood o f W e call such ends geom etrically finite (G F ). W e see that the G F ends o f A f correspond bijectively t o com ponents o f
d f , and thus to com ponents o f A f/. (W e may think o f A f/ as the lim it o f the surfaces d N r { $ ) as r tends to
o o .) If we fix some t) > 0, we can m odify the topological core A fj., so that d N r, ( f r) n A#' becom es a subset o f th e frontier o f M r - T h a t is the frontier com ponents o f M'T in M ' that correspond to G F ends coincide w ith frontier com ponents o f N n[ Y ) O M '.
T h e geom etrically finite ends, however, might n ot a ccoun t for all the ends o f A f I t may be that an end makes no impression on the discontinuity domain fl, so that A hlfors’ Finiteness Theorem tells us nothing. Such ends were shown to exist by Bers and Maskit [Ber,Mas|, their geom etrically infinite nature being made explicit by Greenberg [Gj. Jorgensen later described m ore concrete exam ples [J|. Thurston (Th2) gives a m ore general m ethod o f construction.
A ll the non-G F ends constructed so far have been "sim ply degenerate" as defined by T hurston |Thl C hapter 9|. A sim ply degenerate end turns out to be ju st a p rod u ct topologically (i.e. homeomorphic to a surface tim es a half-open interval), bu t its geometry is infinite. For exam ple, every neighbourhood o f the end w ill contain infinitely m any closed geodesics. Bonahon and O ta l construct an exam ple o f an end containing closed geodesics o f arbitrarily small length (BoO). There are also exam ples w here lengths o f closed geodesics have a positive lower bound. In the latter case the end has bounded diam eter as one tends to infinity. In general, one m ay say that the volum e o f a simply-degenerate end grows at m ost linearly. This explains why such an end makes no impression on the discontinuity dom a in — G F ends have exponential growth.
If, as in all the exam ples constructed so far, each (non-cuspidal) end is either geometrically finite or sim ply degenerate, we call M geom etrically tame. In this case, M is topologically finite, Le. hom eom orphic t o the interior o f a com pact manifold w ith boundary. M oreover, one can show that the limit set o f such a grou p has either sero o r full 2-dim ensional Lebesgue measure (see [T h l] o r |Bo|) — a property conjectured, by A hlfors, for all finitely-generated discrete groups. T here are exam ples, however, where the limit set has H ausdorff dim ension equal to 2, while still having sero 2 -dim ensional Lebesgue measure [Sul2|.
It has been conjectured that all finitely-generated discrete groups are geom etrically tame. Bonahon [Bo| has proven this under the hypothesis that for any free-produ ct decom position r S A • B , there is some parab olic in T not conjugate to any elem ent o f A o r B . O ta l has recently claim ed the result for T 9! Z • Z .
W e now restrict attention to the case where all the ends o f M ' = M \ cu sp (A f) are geometrically finite. T h e n , we call M "geom etrically finite” . In this case, we can assume that each end o f A f' is bounded by a c om p onen t o f d N n( $ ) , which means that we can take the to p olog ica l core A fj. t o be equal t o Nn($ )C \ M ' = A f,,(P )\ cu sp (A f). In other words, geom etric finiteness says that J V ,(^ )\ cu sp (A f) is com pact. This is more o r less the definition o f geom etric finiteness (G F 4 ) due to T h u rston (T h l C hapter 8) (see Section 2 (G F 4 )). (Taking the ^-neighbourhood o f the co n v e x core allows us to include Fuchsian groups and cyclic loxodrom ic groups in the discussion, without making special qualifications.)
Clearly, N n[ f ) meets the bounda ry o f any Z -cu sp in a com p act set. From this we see that the inter section o f A , ( r ) with any Z -cusp has finite volume. (In fact the intersection will be contained in some r-n eigh b ourh ood o f a totally geodesic 2-dimensional cusp — see FIG 0.6.) Since each Z <9 Z -cusp has finite volum e, w e arrive at T hurston’ s second definition o f geom etric finiteness (G F 5 ), namely that Nn( f ) should have finite volum e. (F or the definition G F 4 , it is enough t o insist that ? \ c u s p (A f) be com pact. For G F 5, however, it is essential to take some uniform neighbourhood o f as the exam ple o f an infinitely generated Fuchsian grou p shows.)
If M had no cusps, on e sees that A // = f l / T would g iv e a com pactification o f Af to W<;. In the general case, the topological ends o f M o correspond precisely to the cusps. In fact, each end o f M o has a neighbourhood isometric to one o f tw o standard types — "c u s p tori" and "cu sp cylinders". C usp tori are the same as Z © Z -cusps, whereas a cu sp cylinder is an enlargem ent o f a Z -cu sp t o include a portion o f A f/ (F IG 0.7). This description o f geom etric finiteness (G F l) is due to M arden (Mar|.
A fourth description (G F 2 ), due to Beardon and Maskit |BeaM], dem ands that the limit set should
b e a union o f (w hat we call here) “conica l limit points* and “b ou n ded parabolic fixed points*. These will b e defined in Section 2 (G F 2 ). T h e notion o f a conica l limit p oin t (also called a “ radial lim it point* or “ approxim a tion point*) originates in [H|, and has proven useful t o th e study o f dynam ics o n limit sets.
Finally, the original and simplest definition o f geometric finiteness (G F 3 ) dem ands that T should possess a finite-sided convex fundamental polyhedron. This hypothesis w as introduced b y A hlfors [Ah2], where he sh ow ed th a t the lim it set o f such a grou p must have either sero o r full Lebesgue m< » • « " » h;
It h as been known for som e tim e, from the references already cited, that these five definitions are all equivalent in dim ension 3. G eom etrically finite groups o c cu r frequently as the simplest exam ples o f 3-dim en sion a l hyperbolic groups. It is conjectured that they contain an open dense set o f the space o f all finitely-generated discrete groups, given the appropriate topology (see (Sul5|). T h e hypothesis o f geom etrical finiteness has often been used in the study o f the dynam ics on lim it sets. Sullivan, for exam ples showed that th e lim it set o f a geom etrically finite group is either the whole sphere H j , or else has Hausdorff dim ension s tr ictly less than 2 [Sul3|.
0 . 4 . H i g h e r D im e n s io n s .
A natural question to ask is how one should define geom etric finiteness in dim ensions greater than 3. M o st authors have taken geom etrical finiteness in this case to mean that ‘ he group should possess a finite sid ed c o n v e x fundamental p olyhedron — a direct generalisation o f Ahlfors’ original definition. However, in dim ension 4 and higher, this definition becom es more restrictive than the o b viou s generalisations o f the o th e r fou r definitions. It seems that these other definitions give rise t o a more natural notion o f geom etric finiteness w hich we aim t o elucidate in this work. A ll the applications o f the traditional geom etrical finiteness h yp oth esis seem to b e valid for this slightly more general notion.
T h e question o f defining geom etric finiteness in higher dim ensions has also been considered by A panasov [A p l ,A p 2 ], as well as by W eilenberg |We] and Tuk ia |Tul). In [Tu2], Tukia generalises, t o dim ension n , Su llivan ’ s result ab out the Hausdorff dimension o f th e limit set. T hus, the lim it set o f a G F group is either eq u a l to H " , o r else has Hausdorff dimension less than n — 1.
1 . T h e M a r g u l ia L e m m a a n d B ie b e r b a c h T h e o r e m .
In this section w e shall b e discussing results related to the Margulis Lemm a and B ieberbach Theorem s. O n e form o f the Margulis Lemm a says the follow ing. Given an y positive integer n , w e can find some c (n ) > 0 w ith the following property. Let (Jf, d) be any sim ply connected Riem annian n-m anifold, all o f w h ose sectional curvatures lie in the closed interval [0, lj. Let I* be any discrete group o f isometries acting on X , a n d x 6 X be any point. Let r , ( x ) be the group generated by those elements o f q 6 T such that * ) < « (n ). In sym bols, r , ( x ) = ( 7 € T | d (q (x ), x ) < < (n )). T h en , r , ( x ) is virtually nilpotent (Le. it con ta in s a nilpotent subgroup o f finite index). M oreover, the index o f the nilpotent subgroup in r « (x ) can b e b o u n d e d by some i/(n ) depending on ly on n. W e say that groups o f the form r « ( x ) are uniformly virtually
nilpotent.
A p r o o f o f this result m ay be found in (BaGSj. In this section, w e shall restrict attention to the constant curvatu re cases, namely E™ and H ” , where we can g ive a sim ple p ro o f o f the M argulis Lemm a. Also, in these cases we m ay identify the nilpotent subgroup as being generated b y elements o f sm all rotational part, and it turns o u t always to be abelian. T h is final observation is a consequence o f nilpotency, rather than discreteness, so w e begin with a discussion o f n ilpotent groups o f isometries in the geom etries S n , E " and H " . We shall p rov e th a t nilpotent subgroups o f I s o m s ’*, S i m E " , and Isorn H " are uniform ly virtually abelian. This fact seem s t o b e well known, though I know o f no explicit reference. However all the essential ingredients may b e fou nd in (Th2, Chapter 4j. W e shall go on to show how nilpotent groups arise ou t o f discrete isometry grou p s. In the course o f the discussion we deduce som e o f the classical B ieberbach Theorem s. These results
are also described in (Th2, Chapter 4j and (Wo).
1 . 1 . N i l p o t e n t im p lie s V ir t u a l l y A b e lia n .
L et S'*, E ” and H B denote the unit n-sphere, euclidean n -spa ce and hyperbolic n -spa ce respectively, w ith m etrics d,,,*, d ,ue, and d/,vp. W e shall om it the suffices where there can be no confusion. Let Isom X d e n o te th e entire group o f isometries o f X , and Sim E " be the grou p o f euclidean similarities. Throughout, w e use th e convention o n com m utators that [x,y| = x y x ~ ty ~ l .
W e shall deal w ith the three geometries in turn.
1 .1 (1 ). S p h e r ic a l G e o m e t r y .
L et
I f( S " ) = ( l £ Isom S " | d(~tx,x) < x / 2 fo r a ll * G S“ } .
If w e th in k o f Isom S " acting on E n + l, this says that 7 lies in U (S " ) if it moves each ve cto r through an acute angle. In other w ords, (7v ,v ) > 0 for each non-trivial v ector v G E B+1, where ( , ) is the inner product
defining the metric on E n+1.
L e t 76 Isom S ". B y com plexifying, we can extend 7 to act o n C n+1. Now, 7 preserves the standard herm itia n form on C B + l, Le. the form that restricts to the inner produ ct on E " +1. W e also use ( , ) to d e n o te th is hermitian form .
N o w , let v G C" + 1 b e any non-trivial com plex vector. W rite v = * + i y , with x ,y G E n + I. Then,
Re(7v , w ) - ( 7 * , * ) + < i y ,y ).
I f 7 € i / ( S " ) , both the term s on the right hand side are non-negative, and at least one is s tr ictly positive. It follow s th a t 7 lies in U (S " ) if and only if R e(7V, v) > 0 for each non-trivial v G C " +1.
W e can now prove:
L e p a m a 1 .1 .1 : Let 0 G t f (S " ) and a G Is o m S ". I f a com m utes with [a,0\ , then a com m ute* with 0 .
P r o o f : Com plexifying, we imagine a and 0 acting o n C " +1. W e see that a commutes w ith 0 ~ l a 0 , so that they are sim ultaneously diagonalisable. Let V b e an eigenspace o f or. T h en 0 V is an eigenspace o f 0 a 0 ~ l . If V yi p V , then V must intersect non-trivially som e other eigenspace V ' o f 0 a 0 ~ l , orth ogon a l to 0 V . Let " € V n V ' be non-sero. T h en 0 v lies in f)V , s o that { 0 v , v) = 0. However, since 0 lies in l f ( S n ), the discussion im m ed ia tely prior to the lem m a tells us that R e(^ v, v) > 0. This contradiction means that 0 V — V . Since
V was an arbitrary eigenspace o f a , we deduce that a and 0 are simultaneously diagonalisable, and hence
c om m u te. ❖
C o r o l l a r y 1 .1 .2 : / / T C Iso m S " it nilpotent, then ( m i f ( S " ) ) is abelian.
P r o o f : L et a and 6 lie in r n l f ( S " ) . By a ‘ nested chain o f com m utators* in a and 6, w e mean an expression o f t h e fo r m d — (e*, |e2, . . . [eB, c n+i| ...]) , where each ct is either a o r b. We take J to be o f m axim al length, n , su ch th a t d ft 1. This means that d com m utes with both a and b. It follows that [c2, . . . [ c „ , e « + i| .. -| c o m m u te s with d. A pplying Lemm a 1.1.1, with a = (ca, . . . [ c „ , c n + i ) . . . ) and 0 = Cj, we ded u ce that a and
0 c o m m u te , so that d = 1. W e have contradicted the assum ption that n > 1, and to a m ust com m ute with
O
C o r o l l a r y 1 .1 .3 : Nilpotent subgroup» o f Isom S'* are uniform ly virtually abelian.
1.1(11). E u c li d e a n G e o m e t r y .
T o prepare for the h yperbolic case, it will be useful to consider the g roup Sim E " o f euclidean similarities. Let ♦ be th e set o f parallel classes o f (semi-in finite) geodesic rays in E " . We shall em bed as the unit (n — l)-s p h e r e in an inner-product space V (E n), which we can imagine as euclidean space with a preferred basepoint. There is an obvious bijective correspondence between r-dim ensional subspaces o f K (E B) , and foliations o f E " by parallel r-planes.
T h e g ro u p Sim E " acts isom etrically on 4 , so identifying 4 with S'*- 1 gives us a homomorphism
rot : S iin E " — ► Isom s'*- 1 .
W e call r o t 7 the rotational part o f 7. We define
U (E n) = { 76 Sim E " | rot 7 € i f (S'*- 1 ) } .
Note that if w e embed E m as a plane in E n, then i / ( E ”*) m ay be obtained by intersecting U (E n) with the stabiliser o f this plane. This observation will allow us to use induction over dimension. Given 7 e S im E ” , we shall w rite
min 7 = ( * e E n | d (x , 7 1) is m inim al).
T hen, min 7 is a plane in E " on which 7 acts either trivially o r b y translation. O f course, min 7 may consist o f ju st a single fixed point.
T h e o r e m 1 .1 .4 : //I * C Sim E ” is nUpotent, then
(m
l / ( E " ) ) is abelian.W e shall begin with a lemma.
L ecn m a 1 .1 .5 : Let I* be an abelian subgroup o f S im E '*. L et
r(r)
= Q ^ p i n i n y . Then,r(r)
is a non-em pty, I*-invariant plane, o n which T acts by translations.P r o o f : If T is already a translation group, then r ( r ) = E n , and w e are done. Otherwise, choose any 7 € I' which is n ot a translation. T hen, min 7 is a proper subspace, and since I* is abelian, it is I'-invariant. T he result now follow s by induction on dim ension.
o
In fact, our plane r ( r ) has a natural foliation by (in general) smaller T-invariant planes, namely the set o f m inim al T-invariant planes. T h a t is to say, each leaf is obtain ed as the affine span o f som e T-orbit. This foliation determ ines a subspace W t o f V '(E "), by taking the set o f geodesic rays lying in any o n e leaf. Now,
W i lies in a larger subspace W ' o f V (E n) , determined by r ( r ) itself. Let W2 be the orthogonal com plem ent o f W i in W ', and W3 be the orthogona l com plem ent o f W ' in V [ E n). T his gives us a c anonical decom position
V (E H) = W i © W2 © W3. Let m i be the dim ension o f W,-. We shall say that the decom position is trivial if
mi =* n fo r s om e i.
If m , — n , then T is a pure translation group, and the directions o f translations span E " . If m a — n, then each p oin t o f E " is a fixed p oin t o f I*, thus I* is trivial. If m3 = n , then T has a unique fixed point in E**. W e are now ready for:
P r o o f o f T h e o r e m 1 . 1 . 4 : Let I* be a nilpotent subgroup o f Sim E n . W e shall assume that 1 'is generated by elem ents o f U (E n), Le. that
r
= ( F n U (E ” )). We want to show that T is abelian.Let Z ( r ) b e the centre o f T. From the preceding discussion, Z (V ) determines a decom position W , ©
W2 © W3 o f P ( E " ) . Since this is canonical, it is respected by the whole group T. T hus T splits as a
is abelian. It then follow « that T itself is abelian. W e need therefore d ea l only with the cases when the decom position is trivial.
S uppose m i = n . This means th a t Z (T ) is a translation group with n o non-em pty proper invariant plane in E " . C onsider any 7 € I\ Since 7 com m utes with everything in Z ( T ), min 7 is Z (r)-in v a ria n t, and hence equ a l to E n . It follow s that 7 is a translation o f E " . Since translations com m ute, T is abelian.
S uppose m j = n. N ow Z (T ) is trivial. Since I* is nilpotent, it is also trivial.
Finally, suppose m3 — n. In this case, Z (T ) has a unique fixed p oin t in E '\ T his point must be
fixed by T , so T can b e regarded as a subgroup o f R + x Is o m S ", where the first com ponent measures the m agnification, and the second, the rotationa l part o f an element. T h e projection into Isom S " is nilpotent and generated by elem ents o f ¿ /(S * ). B y C orollary 1.1.2, this projection is abelian. We deduce that T is abelian.
0
A s in th e spherical case, for any group I*, the index o f ( r n l / ( E n) ) in T is finite, and has a bound dependent o n ly o n n. Thus,
C o r o l l a r y 1 . 1 .0 : Nilpotcnt subgroups o f S im E n art uniform ly virtually abelian.
1.1(111). H y p e r b o l i c G e o m e t r y .
W e shall write H " for the ideal (n — l)-sph ere at infinity o f h yperb olic space H " , and write HJt for the com pactifica tion o f hyperbolic space as H " U H " . By a Möbius transform ation on th e sphere S " , we mean any m a p w hich can be represented as a com position o f inversions in (n — l)-spheres. (W e are allowing M öbius transform ations that reverse o rientation.)
W e m a y represent H " , conform ally, as a hemisphere £ o f S " . Isom H " then consists o f those M öbius transform ations w hich preserve E . Let 7 be a M öbius transformation o f S n , with some fixed point y Since 7 acts conform ally, it induces (after scaling) an isometry o f the unit tangent space ( T i S n) v at y. M oreover, we m ay check that if s is any other fixed poin t o f 7, then the induced isom etries on ( T i S " ) , and (T | S n) , are conju gate. T hus, 7 dete. mines a conju gacy class in Isom S'*, which we shall call rot 7. Since our subset I /(S n) o f Isom S " is invariant under conjugacy, it makes sense to dem and that rot 7 should lie in t /( S " ). R estricting t o Is o m H r>, where all M öbius transformations have fixed points, we may define
U (H " ) = { 7 e Isom H " I r ot 7 C lf(S “ ) } . T h e o r e m l . l . T : I f T C Isom H11 is nilpotent, then ( r n l / ( H ’*)) is abelian.
W e b e g in wish tw o lemmas.
L e m m a 1 . 1 .8 : / / T C Isom IT* is abelian, then fix T , the s et o f points fixed by T , consists o f either one o r two points in H ” , o r else is a subspaee o f H £ (i.e . the closure, in H " , o f a plane in H n).
P r o o f : Let 7 be any non-trivial elem ent o f T. If 7 is parabolic, then its fixed poin t is preserved b y T, so that T has a unique fixed point. If 7 is elliptic, then 6x7 is a proper T-invariant subspace, and we use induction on dim ension. For this, we need to check the 1-dim ensional case. B ut it is easily seen that an abelian g ro u p o f isometries o f the real line must either act trivially, or by translation (thus respecting the two “ ideal” p oin ts), o r else consist o f an involution with a single fixed point. Finally, if 7 is loxodrom ic, then its axis is T-invariant, and we are im m ediately reduced t o the 1-dim ensional case.
0
L e m m a 1 . 1 . 0 : Suppose T C Isom H " is nilpotent, then T has a fixed point in H " .
P r o o f : Let o be the set o f points fixed by the centre Z ( r ) . Let T ' 2 Z (T ) be the subgroup that fixes
o pointwise. Since o is canonical with respect t o T, T ' is norm al in T. T hus T/T' is nilpotent, and acts
effectively o n 0.
Prom L em m a 1.1.8, w e distinguish three poeribilitiea for a. Firstly, if a is a single point o f H " , this point is fixed by T, and w e are done. Secondly, if a is a proper subspace o f H " , we use induction o n dimension. Thus, we m a y assum e th a t we are in the third case, namely that a consists o f precisely tw o points, z and y, in H " . If r / r ' is trivial, we are done. Therefore we may suppose that r / V is an involution. This means that there is som e f 6 T that swaps z and y . Now, each element o f Z (I ) fixes z and y, and com m utes with 7. W e see that Z ( r ) m ust fix pointwise the geodesic joining z and y. T h is contradicts the definition o f a as
fixZ(r).
0
P r o o f o f T h e o r e m 1 . 1 .T : By Lemm a 1.1.9, ( r n l / ( H " ) ) fixes som e point, z , o f H £ . If z e H n , we are reduced to the spherical case, and i f z e H ’/ , we are reduced t o th e case o f euclidean similarities. We observe that o u r definitions o f the rotational part o f an isometry (o r sim ilarity) are in agreement, so that the theorem follow s fro m Corollary 1.1.2, and T heorem 1.1.4.
0
For c om pleteness, w e state:
C o r o ll a r y 1 . 1 .1 0 : Nilpotent subgroups o f Isom H " arc uniformly virtually abelian.
P r o o f : If r C Isom H n is nilpotent, we need that [T : ( r n i / ( H n))| is uniform ly bounded. B ut by Lemma 1.1.8, T has a fixed p o in t in H £ , so the result follow s from the spherical and euclidean (similarity) cases. O
Note that a ll th e abelian subgroups constructed in this section are normal, since the neighbourhoods
U (S " ), t / ( E " ) a n d i / ( H n ) are all conjugacy invariant.
1 .2 . D is c r e t e S u b g r o u p s .
In this s ection, w e describe how nilpotent groups occur naturally w hen considering discrete group actions. Let g be a Lie grou p, and let | | be any sm ooth norm on G , for exam ple, distance from the identity in some Riemannian m etric. For any g, h € G , sufficiently near the identity, we will have |jy,Aj| < C|y||A|, for some constant C . T h u s, we can find a bounded symmetric neighbourhood, 0 ( G ) o f the identity in G such that whenever g, h G 0 ( G ) , we have [g, A) e O (G ) and |[y,A]| < jy|/2.
L e m m a 1 .2 .1 : IJ T i t a discrete subgroup o f G , then ( r n O ( C ) ) is nilpotent.
P r o o f : T h e elem ents o f T have norms bounded below by some number e > 0, and the elements o f O (C ) have norm s b ou n d ed above by som e number k. If m is any integer greater than log3(fc /c ), we see that any m -fold c o m m u ta tor in elements o f m O ( C ) will be trivial. B y repeated application o f the identity [zy , s| *■ (z , |y, z)][y, s|(z, *|, we deduce that any m -fold c om m utator in ( r n O ( G ) ) is trivial. Thus, ( r n O ( Q ) ) is nilpotent.
0
The follow ing lem m a is a modified version o f one to be found in (Th2|. It is relevant to o u r discussion o f the M argulis Lem m a. First, we introduce some notation. Given a subset X o f the Lie group G , we write
X r for those g e G expressible as words o f length r, in elements o f X together with their inverses (in X - 1 ),
i.e., inductively, X1 — X U ( l ) U X “ 1, X r = X r - , X * . If T is a subgroup o f G , we write I*x for (I* n X ) .
L e m m a 1 .2 .2 : L et G be a Lie group, with W a neighbourhood o f the identify. Let K t , i e N be a sequence o f symmetric neighbourhoods o f the identity. Suppose K1 is com pact, and ( Kt )4 Q K t fo r each i. Then,
P r o o f : Let V b e a neighbourhood o f I with V ' P C W . Sine« K i ia com pact, there ia an upper bound,
k, o n the num ber o f right tranalatea V g ,g e K t , o f V , that we can pack diajointly in to G . Let N - k + 1.
Suppose that T < G ia discrete. Let = 1 , . . . , p ) be a d iajoint packing with a< e I V W n K i , and p maximal. Note that p < k . W rite I V = ( I V ,, n W ). W e claim that (IV a ,| « = 1 , . . . , p } includes a com plete set o f coaets for I V in VK h , so that (I* * , : I V ) < N , as required.
T o aee this, consider VN h with h 6 TKm. W rite K - I ] L i 9i, with ft e T n K N . If / > k + I, consider the collection { V h j \ j = 1, . . . , * + 1 }, where h ,■ = f l i . i f t . so that A,- e ( Kn) N C K t . These sets cannot all
be disjoint. T h u s, we can write A = a0~i, with a fi e K t and V a n V a f i ? 0. N ow , a 0 a ~ l € V ~ l V C W , so
a p a ~ l € I V T hus, I V A — I V ( a0a - , ) a7 = I V A ', where A' = 0 7. W e have reduced the word-length o f A, so, by induction, I V A — I V A " , with A " e K i . But then, P A " n P o t- j i 0, for som e a ,, so that A" a T l e W , and r « A " = I V a*. Hence, I V A = rjy a i.
❖
W e again consider the three geometries in turn.
l . 3 (1 ). S p h e r ic a l G e o m e t r y .
W e write l/o (S n) for 0 ( I s o m S " ) , the neighbourhood o f the identity d efined at the beginning o f Section 1.2. Since this set m ay be chosen t o be arbitrarily small, we may suppose that i /0(S " ) C l /( S " ). W e m ay also suppose that if0(S " ) is conjugacy invariant. Now if T is a discrete subgroup o f Is o m S ", then (I*f! i /o (S " )) is nilpotent by Lem m a 1.2.1, and thus abelian by C orollary 1.1.2. It is easily checked that ( r n l /o (S “ )) has a finite index in I*, w hich is bounded as T varies. Thus w e have:
L e m m a 1 .2 .3 ( J o r d a n L e m m a ) : Discrete subgroups o /I s o m S n ore uniform ly virtually abelian.
1.2(11). E u c li d e a n G e o m e t r y .
We can assum e that O (ls o m E n) has the form
0 (I s o m E n) — ( 7 6 Isom E " | ¿ (7 0, a) < e and rot 7 e U\ )
where e > 0, a is som e poin t o f E n , and Ui is some neighbourhood o f the identity in Iso m S "- * that is contained in £/(S " - ‘ ). For notational convenience we shall identify Ui with the set l /0( S " - ‘ ) o f the Jordan Lemma. W e set
ff0( E " ) =r { r G Isom E " | rot 7 e l /0( S " ) }.
P r o p o s it io n 1 .2 .4 : Suppose that r is a discrete subgroup o f Isom E " ; then ( m t /0( E " ) ) is abelian. P r o o f : T o begin with, w e d o not know that (T D l /o ( E " ) ) is finitely generated, so we proceed as follows. Let D r = ( 7 G I s r m E " | d (-ja ,a) < r e ). Let g , be the dilation o f magnification r a b ou t a. Considering r r - ( r n U0{E " ) n D r ) , w e see that
ft" 1 r,ft = (ft *rft n t/o(E~) n 0,) - ( f t - T f t n O ( I s o m E " ) )
which is nilpotent (L em m a 1.2.1) and hence abelian (T heorem 1.1.4). Thus, r , is abelian for all r, and so ( r n l /0( E " ) ) - U , r r is abelian.
0
Again, it is easily seen that (I* n t /„ (E n)) has bounded index in T, so w e have:
T h e o r e m 1 .2 .5 ( D l e b e r b a c b ) s Discrete subgroups o f I s o m E " are uniformly virtually abelian. ❖
Note that since l /o ( S " ) is conjugacy invariant in Isom S " , the abelian subgroups we produce in this way will be normal. W e shall w rite i/(n ) for the b ou n d on their index.
W e can say a little m ore about the structure o f discrete euclidean groups:
P r o p o s i t i o n 1 .2 .0 i Suppose T acts properly discontinuously on E n . Then, there ie a plane /a C E n,
preserved by T , with p / T com pact. M oreover, any two such subspaces are parallel, and the action o f T com m utes with the perpendicular translation between them.
P r o o f : If T preserves each o f two planes ri a n d r2, then it preserves f i n r j. It therefore makes sense to speak o f a r-invariant plane p ft 0 being m inim al.
Let p i and p2 b e tw o such minim al planes. Let A (p ^ .p ,) = { x G p< | ci[x, p .) — d (p < ,p y )} C p,-. T preserves A (p,, p , ) . Hence, b y minimality, A (p ,, p , ) = p^. It follows easily that p i and pa must be parallel.
G iven any tw o parallel planes in E n , there is a unique perpendicular translation mapping one to the other. Any isom etry th a t preserves these tw o planes must com m ute with this translation. It follow s that the action o f T on E " m ust com m ute with th e perpendicular translation sending p i t o p 3.
It now remains to show that if T acts m inim ally on E " , then it is cocom p a ct. From the Bieberbach theorem , and the discussion o f abelian groups in Section l .l ( i i ) , we can find a norm al abelian subgroup T', o f finite index in T , a n d a pla ne r < E n, on w hich T ' acts as a coco m p a ct translation group. There are finitely many images, { r » ... r * }, o f r under I*. each preserved by T '. Since a cocom p a ct action is minim al, it follow s that the n are all parallel. W e m ay now find r ', parallel to r , which represents the centre o f mass o f th e u in any transverse plane.
T
preserves r ', so, by minim ality, r ' =E"
= r . Hence,En/ r
is com pact. ❖As in the earlier d iscussion o f the abelian case (Section l .l ( i i ) ) , it is easily seen that the set o f m inim al planes in E " form a folia tio n o f a larger, can on ica l subspace.
1.2(111). H y p e r b o l i c G e o m e t r y .
G iven x G
Hn,
w e w rite/• ( f ) = (t G Isom H " | d (-jr, * ) < « } .
Let d\ b e any Riem annian m etric on the unit t a n ; -nt bundle T i H n o f H n, invariant under the action o f Isom H " . Given x G H " , w e write
! ,'( * ) = { i € Isom H " | di('r«7, 0) < e for each unit ve cto r if based at x } .
If r is a subgroup o f Isom H " , we write
r . ( . ) - < r n / . ( * ) )
r;(*) = <rn /;(,)>.
N ow w e m ay suppose th a t O ( l s o m l l n) has the form I f l x ) for some t i > 0 and x G H " . We also assume that J ,',(x ) S C /(H "). W e n ow have:
P r o p o a it io n 1 .2 .7 s I f T is a discrete subgroup o f Isom H " , (Asa TJt (x ) =■ ( m / . ' j x ) ) is abelian.
Note that, by h om ogen eity, this remains true if we fix <i, and choose x arbitrarily.
W e next show that for sm all «, Tt (x ) is virtually abelian. T o this end, w e take J€', ( x ) to be the set W o f Lem m a 1.2.2, and th e sets K , to be 7 i /r (x ). T h e lemma now tells us that, for some Af > 0,
|r,(n )(s) : ( r (( „ ) ( x ) n / ¿ ( x ) ) ] < Af,
where « (n ) - \/N. T hus,
| r . , a ) ( . ) : r . (« , ( * ) n r ; , ( * ) ) < s .
F or notational convenience, w e shall assume that N < i/(n ), the constant o f the Bieberbach Theorem , and that we have chosen the m etric on T i H " so that <1 = <(n). W e c a li« (n ) the M argulit constant. In summary,
T h e o r e m 1 .2 .8 ( M a r g u l i s L e m m a ) : For all n , there exist <(n) > 0 and i/(n ) € N such that i f T i t any
diecrete subgroup o f Iso m il™ , and * 6 H n, then r , ( n)(x ) has an abelian tubgroup ( ~ r j j n)(x ) n r , | „ | (i )J o f
index at most i/(n).
Note that if 0 < c < c (n ) , then r , ( x )
n
r j ( nj(x ) has index at most i/(n ) inr((s).
B y intersecting all conjugate subgroups to I \ ( x ) D rj|B)(x ), we see that I*( (x ) contains a norm al abelian subgroup o f b ou n ded index, where the bound is independent o f the choice o f discrete subgroup I\2 . F i v e D e fin it io n s o f G e o m e t r i c a l F in ite n e s s .
In this section, we sh a ll give details o f the five definitions o f geom etrical finiteness that we in tend to use. First, we clarify a few poin ts o f terminology, and notation.
By a (discrete) parabolic group o f hyperbolic isometries, G , we mean a discrete group which fixes a unique point in H " , i.e. (~)1( 6*1= ( p ), where p e H " . In this case, G m ust contain at least one p arabolic with fixed point p. Since n o loxodom ic can share a fixed poin t w ith a parabolic in any discrete group, we see that G consists entirely o f parabolics and elliptics. We may represent H " using the upper half-space m odel R " with p — 00. It then follow s that G acts as a group o f euclidean isometries o f 3 R " = E n _ l . From Proposition 1.2.6, w e know th a t G preserves some plane in d R " whose quotient by G is com pact. M oreover, any two such planes are p arallel. W e shall write <7/ for som e choice o f such plane, and write <r for the v ertica l euclidean half-space w ith <r n d R " = <r/. Thus, <r is the hyperbolic subspace spanned by <7/ and p. N ow , if r is any discrete su bgrou p o f H " , and 7 e T is any parabolic, with fixed poin t p, then the stabiliser o f p is T will be a parabolic su b g rou p . We call p a parabolic fixed p oint, which we abbreviate to p.f.p.
G F 1 Let r be a discrete grou p . In Section 0.2, we defined M e as the quotient, b y T, o f hyperbolic space together with the discontinuity domain, that is M e =
(H"Uil)/r.
T hus M o — MU
M i, where M =H "/r
is a com plete h yperbolic m anifold, and M/ = 0 /1 ? consists o f ‘ ideal points” o f M . W here there is more than on e group in question, w e shall be specific by writing
M(r), Af/(r)
and A fc (F ).Suppose that T and P are two discrete groups. Suppose c and s' are (topological) ends o f Af0- ( r ) and
M ‘C ( V ) respectively. W e sa y that e and e ' are equivalent if they adm it isom etric neighbouroods. (Here, w e
use the term ‘ isometric* loosely, in that the orbifolds in question may contain ideal points. In saying that tw o such orbifolds are isom etric, we mean that there is an isometry o f the metric parts which extends t o a homeomorphism on th e ideal points.) Note that if T ' is a parabolic group, then A f c ( r ') has precisely one end (see the discussion b e lo w ).
D e fin it io n 1 : T u G F I i f Afc ( r ) hat finitely many ends, and each in ch end it equivalent to the end o f
the quotient o f a parabolic group.
It will be convenient t o give this definition a moi • concrete form ulation in terms o f the structure o f parabolic groups. L et r p b e such a group, fixing p = x> in the upper half-space model, R " . Let
a be a r-invariant vertica l plane with
<7//rr
com pact. Let C (r ) = { * € R " U 3 R " | d ,M (x,<ri) > r ) . T hus, R ’J u 3 R " \ C ( r ) is an open uniform r neighbourhood o f <r/ in the euclidean metric on R " U 3 R " . Since Tp preserves the euclidean metric, the constuction is r p-equivariant, so we may form the quotient ¿ ( r ) = C (r)/ T p. Since «7/ is com pact, ( R " u 3 R " \ C ( r ) ) / r , = * f c ( r , ) \ ( J ( r ) is relatively com pact inM o ( r p). If we take a seq uence (r „ ) tending to infinity, then the sets M c ( r p) \<?(rH) give a relatively
com pact exhaustion o f M o ( T p). Since each & (rn) is connected, we see that M a (T p) has precisely on e end. T h e sets ¿ ( r n ) give a n eighbourhood base for the end. Given any r, we call C [r ) a itandard parabolic region (F IG 2.1), and the q u otien t 6(r ), a itandard eutp (F IG 2.2).
W e may now say that T is G F 1 if and only if w e can write M e = l^ U ( (j£ ) > with A co m p a c t, t finite, and each <? G t (isometric to) a standard cu»p (F IG 2.3 ). We shall see (Cha pter 3, G F 4 => G F 2 ) that the cusps 6 are in bijective correspondence to the orbits o f parabolic fixed poin ts o f I*.
W e w rite N for the lift o f f t t o H £ .
G F 2
Let p be a parabolic fixed p oin t (p.f.p.) of the discrete group I*. Let Tp = stabpp — the stabiliser o f p. W e say that P is a bounded p .f.p . (b .p .f.p .) if ( A \ { p ) ) / r p is com pact. Let p = oo in the ip p er half-space model, and let <r/ be a mininmal Tp invariant plane. Then it is not difficult to see that ( A \ { o o } ) / r p is com pact if and only if deuc(y , a/) is bounded as y varies in A \ {o o ). In other words, p is a b .p .f.p . if and only if A (r ) \ { o o ) C Q r = ( x G 3 R 1} | dIac( i , f f j ) < r } for some r (F IG 2.4).
Let y G A(T). We say that y is a conical limit point (c .l.p .) if for som e (an d hence every) geodesic ra y l joining some point o f H n t o y, the o r b it Tl o f l accum ulates somewhere in H -‘ , Le., |{-j G T\-jln K f f)| = o c for som e com pact K C I I " . (T h e term derives from an alternative desription, namely that there should exist a sequence ~in G T , and a p oin t x G H n, with 7„ x tending to y , while remaining a b ou n ded distance from som e geodesic ray — see F IG 2.5.)
D e fin it io n 2 : T is G F t i f e v e ry point o f A it eith er a c.l.p. o r a b.p.f.p.
W e shall see in C h.3 that these tw o classes are, in any discrete grou p, mutually exclusive. In fact, it is shown in [SusS| that, in any discrete group, no p.f.p. can also be a c.L p. It will follow from th e discussion in C hapter 3 G F 4 => G F 2 that in th e special case o f a geometrically finite group, any p.f.p. is necesserily a b.p.f.p., and thus not a c.l.p.
Beardon and M askit |BeaM| give several equivalent definitions o f c.l.p ., including one that makes sense in H " . This gives G F 2 as a definition o f geometric finiteness intrinsic to the action o f T on H " .
Finally, we remark that, for a G F group, the convergence o f orbits under T t o c.Lp.s can b e chosen to be "uniform *. For us, this means th a t the set K , in th e definition, can b e chosen independently o f the poin t y and the ray l. Together with a certain convergence property for the radii o f isometric spheres, this implies that the limit set o f a G F group has either sero o r fu ll spherical Lebesgue measure (see (B e a M .A p I)). A more geom etric proof o f this fact is based on the definition GF5 (see (T h lj).
G F 3
Let T b e a discrete group. W e have said that the hypothesis that T should possess a finite-sided fundamental polyhedron is m ore restrictive than w e w ould like in dim ension 4 onwards. In S ection 4, we give an exam ple to illustrate this p oin t (at least for th e case o f Dirichlet dom ains). However it is possible to m odify the criterion s o that it w orks in all dimensions. T h e idea is that w e should allow ourselves m ore than one polyhedron to c onstitute a fundam ental domain fo r I\ T h e definition is m ost clearly expressed in term s o f w hat w e shall call “ convex cell com p lexes'. A convex cell com plex is cell com plex in w hich all the cells are convex, and hence necessarily polyhedra. It need n ot quite be a C W -co m p le x since w e on ly attach cells along their relative boundaries in hyperbolic space. T h u s a finite com plex is com plete, bu t n ot in general com pact. W e give a m ore form al description below.
Let A be a subset o f E " . W e call A an open (c o n v e x ) cell if any tw o distinct points o f A lie in the interior o f some geodesic segm ent contained entirely in A . Note that by demanding that the tw o points be distinct, we allow any one-point s et as an open c e ll W e see that the prop erty o f being an open cell is closed under taking finite intersections.
D e fin it io n : A collection A o f subsets o f E " is "convex cell complex" if:
(1 ) each elem ent o f A is an open cell, ( t ) the sets o f A are all disjoint, (3 ) the collection A is locally finite, (4 ) I M = E » ,
( 5 ) I f A , B G A and B r A j t $ , then B Q A.
Let A be such a cell c om plex, and let B G A. Suppose that x and y are two points o f B , and suppose that x G A for some cell A G A. T h e n , from (5), we see that y G A . T hus, { A G A \ x G A ) is in dependent
o f the choice o f x 6 B . In particular, from the local finiteneee o f A, (2), we see that an y cell o f A m eets th e closures o f only finitely m any oth er cells.
Now, given two cell com plexes A and B, we call B a subdivision o f A if each B G B is a subset o f some A € A. Any two cell com p lexes Ai and A3, have a natural com m on subdivision, namely ( A i, A3)
= { A i n A t \ A i € A i , A i € A a ) . In fact (A t, A ]) is m inim al with respect to subdivision — if B is a subdivision o f b oth Ai and A3, then it is a subdivision o f (A\, A3). We also rem ark th a t intersecting a cell
com plex with an affine subspace o f E n gives a cell com plex in that subspace.
A ll the above properties are easily verified from the definition above. However, to make the an a logy with C W -com plexes more explicit, w e offer a slightly different description o f convex cell com plexes as follow s. Suppose that A C E ” is a c o n v e x set. W e write (A ) for the affine span o f A , i.e. th e smallest su bsp a ce o f E " containing A . We m ay define the dim ension o f A , dim A , t o b e equal to the dim ension o f (A ). W e also define ri A and rb A to be, respectively, the relative interior and the relative b ou n da ry o f A in (A ). N o te that ri A is always nonempty, provided A is nonempty. M oreover, it is not difficult t o see that A is an o p e n cell if and only if r iA = A . B y an open i-e tll, we mean an o p en cell o f dim ension ».
Let A b e a collection o f c onvex cells o f E n . We w rite A ' for set o f all t-cells in A . T h e r-skeleton , K T, o f
A is the union o f all t-cells with t < r , Le. K T = U I (L M ')- W e now claim that, if w e know that A satisfies
properties ( l ) - ( 4 ) , then property (5 ) is equivalent to the following: (S ') If A € A ', then r b A £ K r~ x.
T o see (5 ') => (S), it is enough to not that if one open cell lies in the relative boundary o f an oth er, then its dim ension must be strictly less. T o see (5) => (S ') is a little more com plica ted. Suppose th a t A satisfies ( l ) - ( 4 ) and (5), and let A 6 A '. Using (for exam ple) a measure-theoretic argum ent, we see th a t (U Ar_ 1 ) n r b A is a dense subset o f d A . Suppose that B € Ar _ 1 intersects rb A . If B is not a subset o f r b A , then there is some point x in th e relative boundary o f A
n
B in B . By considering a neighbourhood o f x inr b A , we see that x € r b C for som e C € Ar ~ l , different from B . But by (S '), w e have that r b C £ K T~ a. This contradiction tells us that B £ r b A . In other words, r b A is a union o f closures o f (r — l)-ce lls . W e have deduced property (5) in th e case where dim A - d im B = 1. W'c new use induction over dim A — d im B . Let D € A be an t-cell intersecting r b A . From (5 '), we know that t < r — 1. If « = r — 1, we are d on e. If t < r — 1, then D intersects r b £ fo r som e (r — l)-c e ll E £ r b A . B y the induction hypothesis, 0 £ r b £ . W e see that D £ rb A . Thus w e have show n the equivalence o f the tw o descriptions o f c o n v e x cell com plexes.
Now , let A be a convex cell com plex, and let U — A " b e the collection o f top-dim ensional cells. W e claim that U is characterised by th e following properties.
(a) U is a collection o f open c onvex subsets o f E n . (b ) U is locally finite.
(c ) T h e closures o f all the sets in U cover E n . (d) T h e elements o f U are disjoint.
In fact, if we are given such a c ollection , we m ay recover a c onvex cell com plex as follow s. Given x € E " , w e write P ( x ) - {U e U | x 6 0 ) . L et A (x ) *- (* e E " | P (y ) = P (x )> , and let A (U ) - ( A ( x ) | x € E " ) . T h e n , we claim that A(U ) is a convex cell com plex with A (U )H — U- T h e only property that is n ot im m ediate is property ( 1 ), namely that each A (x ) is an open celL For this, it is enough to check that if y and s are distin c t points in E " with P (y ) = P ( y ) , then y and s lie in the interior o f a line segment i, w ith P (u ) = P ( y ) fo r all u € /. In fact, if T is the translation o f E " sending y to *, then it is not difficult to see that, for som e « > 0, we w ill have J V ,(* )n t/ — T (N ,[y )r \ U ), for any U € U . (AT, denotes a uniform «-n eigh bou rh ood.) T h is m eans that w e can find such a line segm ent with a neighbourhood on which the sets o f U are a cartesian p ro d u ct. W e deduce that A(U ) is a cell c om p lex, which has U as its collection o f top-dim ensional cells. M oreover,
A(U) is m inim al with respect to subdivision. Thus, our original A is a subdivision o f A (U ).
W e state a refinement o f th e a b ov e result.
P r o p o s i t i o n 2 .1 : L et A he a c o n v ex cell complex. L et U he a locally finite collection o f open r-cells, w hose
closures cover the r-skeleton o f A . Suppose that each U € U is a subset o f som e A € A '. Then there is a natural subdivision, B, o f A , such that BT * U and B ' = A* f o r i » f + l ...s .
T h e propoeition may be proven by similar arguments to those given above. In fa ct, our discussion d ea lt with the special case when A = { E n } and r *» n. W e shall n ot give details here, since it is not central t o the
