On geodesic homotopies of controlled width and conjugacies in isometry groups
aa r X i v : . [ m a t h . G T ] O c t On geodesic homotopies of controlled width andconjugacies in isometry groups
Gerasim Kokarev
School of Mathematics, The University of EdinburghKing’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UKEmail:
Abstract
We give an analytical proof of the Poincar´e-type inequalities for widths of geodesichomotopies between equivariant maps valued in Hadamard metric spaces. As an appli-cation we obtain a linear bound for the length of an element conjugating two finite listsin a group acting on an Hadamard space.
Mathematics Subject Classification (2000):
0. Introduction
Let M and M ′ be smooth Riemannian manifolds without boundary. For a smooth mapping u : M → M ′ by E ( u ) we denote its energy E ( u ) = Z M k du ( x ) k dVol ( x ) , (0.1)where the norm of the linear operator du ( x ) : T x M → T u ( x ) M ′ is induced by the Riemannianmetrics on M and M ′ . Let u and v be smooth homotopic mappings of M to M ′ and H ( s , · ) be a smooth homotopy between them. The L -width W ( H ) of H is defined as the L -normof the function ℓ H ( x ) = the length of the curve s H ( s , x ) , x ∈ M . (0.2)A smooth homotopy H ( s , x ) is called geodesic if for each x ∈ M the track curve s H ( s , x ) is a geodesic.In [6, 7] Kappeler, Kuksin, and Schroeder prove the following geometric inequalityfor the L -widths of geodesic homotopies when the target manifold M ′ is non-positivelycurved. Width Inequality I.
Let M and M ′ be compact Riemannian manifolds and suppose that M ′ has non-positive sectional curvature. Let z be a homotopy class of maps of M to M ′ . Thenthere exist constants C ⋆ and C with the following property: any smooth homotopic maps uand v ∈ z can be joined by a geodesic homotopy H whose L -width W ( H ) is controlled bythe energies of u and v, W ( H ) C ⋆ ( E / ( u ) + E / ( v )) + C . (0.3) Moreover, if the sectional curvature of M ′ is strictly negative, the constants C ⋆ and C canbe chosen to be independent of the homotopy class z . G be a finite graph and u : G → M ′ be a smooth map, that is whose restriction to every edge is smooth. The length L ( u ) of u isdefined as the sum of the lengths of the images of the edges. By the L ¥ -width W ¥ ( H ) of ahomotopy H we mean the L ¥ -norm of the length function ℓ H ( x ) , given by (0.2). Width Inequality II.
Let G be a finite graph and M ′ be a compact manifold of non-positive sectional curvature. Let z be a homotopy class of maps G → M ′ . Then there existconstants C ⋆ and C with the following property: any smooth homotopic maps u and v ∈ z can be joined by a geodesic homotopy H such thatW ¥ ( H ) C ⋆ ( L ( u ) + L ( v )) + C . Moreover, if the sectional curvature of M ′ is strictly negative, the constants C ⋆ and C canbe chosen to be independent of the homotopy class z . The purpose of this note is two-fold: firstly, we generalise the width inequalities to theframework of equivariant maps valued in Hadamard spaces. This, in particular, includeswidth inequalities for homotopies between maps into non-compact metric target spaces. Incontrast with the geometric methods in [6, 7] (and also in [2]), we give an analytical proofof the width inequalities via harmonic map theory.Secondly, we use width inequalities for equivariant maps of trees to obtain informaionon algebraic properties of finitely generated groups L acting by isometries on Hadamardspaces. More precisely, under some extra hypotheses, these groups satisfy the followingproperty: given two finite conjugate lists of elements ( a i ) i N and ( b i ) i N in L thereexists g ∈ L with b i = g − a i g such that | g | C ⋆ N (cid:229) i = ( | a i | + | b i | ) + C , where |·| stands for the length d ( · , e ) in the word metric on L . If the group L has a solubleword problem, then the latter estimate yields immediately the solubility of the conjugacyproblem for finite lists in L .
1. Statements and discussion of results
Let M be a compact Riemannian manifold without boundary; we denote by ˜ M its universalcover and by G the fundamental group p ( M ) . Let ( Y , d ) be an Hadamard space; that is acomplete length space of non-positive curvature in the sense of Alexandrov (see Sect. 2 fora precise definition). Denote by r a representation of G in the isometry group of Y . Recallthat a map u : ˜ M → Y is called r - equivariant if u ( g · x ) = r ( g ) · u ( x ) for all x ∈ ˜ M , g ∈ G . r -equivariant maps u and v the real-valued functions d ( u ( x ) , v ( x )) , where x ∈ ˜ M , are in-variant with respect to the domain action and, hence, are defined on the quotient M = ˜ M / G .In particular, the quantity d ( u , v ) = (cid:18) Z M d ( u ( x ) , v ( x )) dVol ( x ) (cid:19) / (1.1)defines a metric on the space of locally L -integrable r -equivariant maps. The latter can bealso regarded as the L -width of a unique geodesic homotopy between r -equivariant maps.If u is a locally Sobolev W , -smooth r -equivariant map, then its energy density measure | du | dVol (see Sect. 2) is also G -invariant and the energy of u is defined as the integral E ( u ) = Z M | du | dVol . (1.2)Recall that the ideal boundary of Y is defined as the set of equivalence classes of asymp-totic geodesic rays, where two rays are asymptotic if they remain at a bounded distancefrom each other. Clearly, any action of G by isometries on Y extends to the action on theideal boundary. Theorem 1.
Let M be a compact Riemannian manifold without boundary and Y be a lo-cally compact Hadamard space. Let G be the fundamental group of M and r : G → Isom ( Y ) be its representation whose image does not fix a point on the ideal boundary of Y . Thenthere exists a constant C ⋆ such that for any r -equivariant locally W , -smooth maps u andv the L -width of a geodesic homotopy H between them satisfies the inequalityW ( H ) C ⋆ ( E / ( u ) + E / ( v )) . (1.3)The proof appears in Sect. 3. The idea is to prove first a similar inequality when oneof the maps is an energy minimiser, and then to use compactness properties of the mod-uli space formed by such minimisers. The former is based on a compactness argument,mimicking the proof of the classical Poincar´e inequality.Below we state a version of Theorem 1 for equivariant maps of trees. First, we introducemore notation. Let G be a finite connected graph without terminals and G be its fundamentalgroup p ( G ) . By T we denote the universal covering tree of G ; the group G acts naturallyon T by the deck transformations. As above the symbol r denotes a representation of G in the isometry group of an Hadamard space Y . For a locally rectifiable r -equivariant map u : T → Y , its length density measure | du | dt (see Sect. 2) is G -invariant and the length of u is defined as the integral L ( u ) = Z G | du | dt . Theorem 2.
Let G be a finite graph and Y be a locally compact Hadamard space. Let G bethe fundamental group of G and r : G → Isom ( Y ) be its representation whose image doesnot fix a point on the ideal boundary of Y . Then there exists a constant C ⋆ such that forany locally rectifiable r -equivariant maps u and v the L ¥ -width of a geodesic homotopybetween them satisfies the inequalityW ¥ ( H ) C ⋆ ( L ( u ) + L ( v )) . (1.4) Example.
Let M ′ be a (not necessarily compact) Riemannian manifold whose sectionalcurvature is negative and bounded away from zero and the injectivity radius is positive. Let r : G → p ( M ′ ) be a homomorphism whose image is neither trivial nor infinite cyclic. Then3he latter does not fix a point on the ideal boundary of the universal cover of M ′ . Indeed,the group r ( G ) is generated by hyperbolic elements (regarded as isometries of the universalcover), see [7, Lem. B.1], and the statement follows from the results in [3, Sect. 6]. Thus, asa particular case, Theorem 2 contains the width inequality for homotopies between mapsfrom G to M ′ ; this is the situtation considered in [7, Th. 0.1]. (Under the hypotheseson the homomorphism r , the homotopy class is neither trivial nor contains a map onto aclosed curve.) The methods in [6, 7] do not seem to yield an analogous L -width inequality(provided by Theorem 1) for non-compact targets when the dimension of the domain isgreater than one.We proceed with width inequalities for representations in co-compact subgroups ofIsom ( Y ) . Recall that an action of a group L on a metric space ( Y , d ) is said to be co-compact if the quotient Y / L is compact. Further, the action of L is said to be proper if foreach y ∈ Y there exists r > { g ∈ L | g · B ( y , r ) ∩ B ( y , r ) = ∅ } is finite. Fora homomorphism r : G → L we denote by Z below the centraliser of the image r ( G ) in L . Theorem 3.
Let M be a compact Riemannian manifold without boundary and Y be alocally compact Hadamard space. Let L be a group acting properly and co-compactly byisometries on Y . Denote by G the fundamental group of M and let r be a homomorphism G → L . Then there are constants C ⋆ and C such that for any r -equivariant locally W , -smooth maps u and v there exists an element h ∈ Z such that the L -width of a geodesichomotopy H between u and h · v satisfies the inequalityW ( H ) C ⋆ ( E / ( u ) + E / ( v )) + C . Theorem 4.
Let G be a finite graph and Y be a locally compact Hadamard space. Let L bea group acting properly and co-compactly by isometries on Y . Denote by G the fundamentalgroup of G and let r be a homomorphism G → L . Then there are constants C ⋆ and C suchthat for any locally rectifiable r -equivariant maps u and v there exists an element h ∈ Zsuch that the L ¥ -width of a geodesic homotopy H between u and h · v satisfies the inequalityW ¥ ( H ) C ⋆ ( L ( u ) + L ( v )) + C . Remark.
If the homomorphism r : G → L in the theorems is trivial, then the second con-stant C is equal to diam ( Y / L ) Vol / M and diam ( Y / L ) in the L - and L ¥ -versions respec-tively. For non-trivial representations of G it can be chosen to be zero. Example.
As a partial case, when the action of L is free, Theorems 3 and 4 above containwidth inequalities for homotopies between continuous W , -smooth maps valued in a com-pact metric space Y / L . The choice of an element h ∈ Z in this setting corresponds to thechoice of the homotopy between maps. Indeed, recall that the fundamental group of thespace formed by continuous maps homotopic to u : M → Y / L is equal to the centraliser ofthe image u ∗ ( p ( M )) in L . Now we describe some applications of the width inequalities to geometric group theory.First, recall that a discrete subgroup L in a Lie group G is called lattice if the quotient G / L carries a finite G -invariant measure. Such a lattice is always finitely generated providedthe group G is semi-simple and has rank >
2; see ref. in [11]. Choose a finite system ofgenerators ( g i ) of L and consider the word metric d ( · , · ) on L associated with the Cayleygraph determined by the generators. Denote by | g | the length d ( g , e ) , the distance betweenan element g and the neutral element e . 4he following statements are essentially consequences of Theorems 2 and 4 and areexplained in Sect. 4. Theorem 5.
Let G be a semi-simple Lie group of rank > all of whose simple factors arenon-compact. Let L be an irreducible lattice in G and ( a i ) i N be a finite list of elementsin L which does not fix a point on the ideal boundary of the associated symmetric space.Then for any conjugate (in L ) list ( b i ) i N any conjugating element g ∈ L , b i = g − a i g,satisfies the inequality | g | C ⋆ N (cid:229) i = ( | a i | + | b i | ) + C , where the constants depend only on the conjugacy class of the lists. In particular, for suchtwo given lists the set of conjugating elements is finite.Remark. An analogous statement holds if L is an irreducible lattice in an almost simple p -adic algebraic Lie group of rank >
2. In this case we consider lists which do not fixpoints on the ideal boundary of the associated Euclidean building.
Example.
When the group G is algebraic, the hypothesis on the finite list ( a i ) is satisfiedif, for example, the elements a i ’s generate a lattice (e.g., the whole group L ) in G . Indeed,by Borel’s density theorem the latter is Zariski dense in G and, hence, does not fix a pointon the ideal boundary of the associated symmetric space.The estimate above yields immediately an algorithm deciding whether a given list ofelements in L is conjugate to the list ( a i ) in the theorem. This is a special case of themore general result due to Grunewald and Segal [5]: the conjugacy problem for finite listsin arithmetic groups is soluble. (Any irreducible lattice in a semi-simple Lie group ofrank > | g | we denote the length d ( g , e ) in the wordmetric. Theorem 6.
Let Y be a locally compact Hadamard space and L be a group acting properlyand co-compactly by isometries on Y . Then for any finite conjugate lists ( a i ) i N and ( b i ) i N of elements in L there exists an element g ∈ L with b i = g − a i g such that | g | C ⋆ N (cid:229) i = ( | a i | + | b i | ) + C , (1.5) where the constants depend only on the conjugacy class of the lists. Further, there existsan algorithm deciding whether two given finite lists of elements in L are conjugate. When the list ( a i ) in the theorem consists of a single element, the solubility of theconjugacy problem is well-known. It is, for example, a consequence of an exponential(compare with our linear) bound for the length of the conjugating element in [1, III. G .1.12].In the context of decision problems it is worth noting that there are finitely presented groupsin which the conjugacy problem for elements is soluble, but the conjugacy problem forfinite lists is not. We refer to [2] for the explicit examples. Finally, mention that in [2]Bridson and Howie prove a closely related linear estimate for the length of the conjugating(two finite lists) element in Gromov hyperbolic groups.5 . Preliminaries We recall some background material on Sobolev spaces of maps valued in a metric space.The details can be found in [10].Let W be a Riemannian domain and ( Y , d ) be an arbitrary metric space. We supposethat W is endowed with a Lebesgue measure dVol induced by the Riemannian volume. Ameasurable map u : W → Y is called locally L -integrable if it has a seperable essentialrange and for which d ( u ( · ) , Q ) is a locally L -integrable function on W for some Q ∈ Y (and, hence, by the triangle inequality for any Q ∈ Y ). If the domain W is bounded, thenthe function d ( u , v ) = (cid:18) Z W d ( u ( x ) , v ( x )) dVol ( x ) (cid:19) / defines a metric on the space of locally L -integrable maps. The latter is complete provided Y is complete.The approximate energy density of a locally L -integrable map u is defined for e > e e ( u )( x ) = Z S e ( x ) d ( u ( x ) , u ( x ′ )) e n + dVol ( x ′ ) , where S e ( x ) denotes the e -sphere centred at x and n stands for the dimension of W . Thefunction e e ( x ) is non-negative and locally L -integrable. Definition.
The energy E ( u ) of a locally L -integrable map u is defined as E ( u ) = sup f (cid:18) lim e → sup Z W f e e ( u ) dVol (cid:19) , where the sup is taken with respect to compactly supported continuous functions whichtake values between 0 and 1. A locally L -integrable map u is called locally W , -smooth if for any relatively compact domain D ⊂ W the energy E ( u | D ) is finite.Due to the results of Korevaar and Schoen [10, Sect. 1] a locally L -integrable map u is locally W , -smooth if and only if there exists a locally L -integrable function e ( u ) such that the measures e e ( u ) dVol converge weakly to the measure e ( u ) dVol as e →
0. Thefunction e ( u ) , also denoted by | du | , is called the energy density of u , and the energy E ( u ) is equal to the total mass R e ( u ) dVol .Now suppose that the domain W is 1-dimensional, that is an interval I = ( a , b ) . For amap u : I → Y one can also define the approximate length density as l e ( u )( t ) = d ( u ( t ) , u ( t + e )) + d ( u ( t ) , u ( t − e )) e , t ∈ I . Then the length of u is defined by the formula similar to that for the energy, L ( u ) = sup f (cid:18) lim e → sup Z I f l e ( u ) dt (cid:19) , where the sup is taken with respect to compactly supported continuous functions. A map u : I → Y is called rectifiable if its length is finite. In this case there exists a length densityfunction (or speed function ) l ( u ) such that the lenght L ( u ) equals R l ( u ) dt .6 .2. Hadamard spaces Recall that an
Hadamard space ( Y , d ) is a complete metric space which satisfies the fol-lowing two hypotheses:(i) Length Space.
For any two points y and y ∈ Y there exists a rectifiable curve g from y to y such that d ( y , y ) = Length ( g ) . We call such a curve g geodesic .(ii) Triangle comparison.
For any three points P , Q , and R in Y and the choices ofgeodesics g PQ , g QR , and g RP connecting the respecting points denote by ¯ P , ¯ Q , and ¯ R the vertices of the (possibly degenerate) Euclidean triangle with side lengths ℓ ( g PQ ) , ℓ ( g QR ) , and ℓ ( g RP ) respectively. Let Q l be a point on the geodesic g QR which is afraction l , 0 l
1, of the distance from Q to R ; d ( Q l , Q ) = l d ( Q , R ) , d ( Q l , R ) = ( − l ) d ( Q , R ) . Denote by ¯ Q l an analogous point on the side ¯ Q ¯ R of the Euclidean triangle. Thetriangle comparison hypothesis says that the metric distance d ( P , Q l ) (from Q l tothe opposite vertex) is bounded above by the Euclidean distance (cid:12)(cid:12) ¯ P − ¯ Q l (cid:12)(cid:12) . Thisinequality can be written in the following form: d PQ l ( − l ) d PQ + l d PR − l ( − l ) d QR . (2.1)It is a direct consequence of the property (ii) above that geodesics in an Hadamard spaceare unique. It is also a consequence of geodesic uniqueness that an Hadamard space hasto be simply-connected [1, II.1]. Examples include symmetric spaces of non-compact typeand Euclidean buildings, simply-connected manifolds of non-positive sectional curvature,Hilbert spaces, simply-connected Euclidean or hyperbolic simplicial complexes satisfyingcertain local link conditions [1, II.5.4]. Another class of examples is provided by the fol-lowing proposition. Proposition 1.
Let M be a compact Riemannian manifold without boundary and ( Y , d ) bean Hadamard space. Let r be a represenation of the fundamental group G = p ( M ) in thegroup of isometries of Y . Then the space of r -equivariant locally L -integrable maps from ˜ M to Y endowed with the metric (1.1) is an Hadamard space.
The proof follows straightforward from the definitions: the geodesics in the new spaceare geodesic homotopies and the triangle comparison hypothesis follows by integration ofrelation (2.1).A useful consequence of the triangle comparison hypothesis is the following quadrilat-eral comparison property due to Reshetnyak [12] (we refer to [10, Cor. 2.1.3] for a proof).
Proposition 2.
Let ( Y , d ) be an Hadamard space and P, Q, R, and S be an ordered sequenceof points in Y . For l , m define P l to be the point which is the fraction l of theway from P to S (on the geodesic g PS ) and Q m to be the point which is the fraction m of theway from Q to R (on the opposite geodesic g QR ). Then for any a , t the followinginequality holds:d P t Q t ( − t ) d PQ + td RS − t ( − t ) (cid:2) a ( d PS − d QR ) + ( − a )( d RS − d PQ ) (cid:3) . (2.2)7etting a to be equal to zero in this inequality, we deduce the convexity of the distancebetween geodesics d P t Q t ( − t ) d PQ + td RS . (2.3)This implies the following energy convexity property. Let u and v be locally W , -smoothmaps from the Riemannian domain W to an Hadamard space ( Y , d ) . Let H ( s , · ) be ageodesic homotopy between u and v ; the point H ( s , x ) is the fraction s of the way from u ( x ) to v ( x ) , where x ∈ W . Then for any s the map H ( s , · ) is locally W , -smooth and forany relatively compact domain D ⊂ W its energy satisfies the inequality E / ( H s ) ( − s ) E / ( u ) + sE / ( v ) . (2.4)Inequality (2.3) also yields the length convexity along geodesic homotopies. More pre-cisely, let u and v be rectifiable paths in ( Y , d ) and let H ( s , · ) be a geodesic homotopybetween them parameterised by the arc-length as above. Then for any s the map H ( s , · ) isrectifiable and its length satisfies the inequality L ( H s ) ( − s ) L ( u ) + sL ( v ) . (2.5)Another consequence of the triangle comparison hypothesis is the existence of the near-est point projection p : Y → A onto a convex subset A . In more detail, if ( Y , d ) is anHadamard space and A is its non-empty closed convex subset, then for any y ∈ Y thereexists a unique point a ∈ A which minimises the distance d ( y , a ) among all points in A ;see [10, Prop. 2.5.4]. Let M be a compact Riemannian manifold without boundary and ( Y , d ) be an Hadamardspace. As above by G we denote the fundamental group of M and by r : G → Isom ( Y ) itsrepresentation in the isometry group of Y . We consider r -equivariant locally W , -smoothmaps u from the universal cover ˜ M to Y . The energy density of such a map u is a G -invariant function on ˜ M , which can be also regarded as a function on the quotient M = ˜ M / G .In particular, by the energy E ( u ) we understand the integral R M e ( u ) dVol . We call a r -equivariant map harmonic if it minimises the energy among all r -equivariant locally W , -smooth maps.The following statement is a straightforward consequence of the energy convexity, for-mula (2.4). We state it as a proposition for the convenience of references. Proposition 3.
Under the hypotheses above, let u and v be two r -equivariant harmonicmaps and H ( s , · ) be a geodesic homotopy between them; the point H ( s , x ) is the fractions of the way from u ( x ) and v ( x ) , where x ∈ ˜ M. Then for each s the map H ( s , · ) is also r -equivariant harmonic and the energy E ( H s ) does not depend on s. We proceed with the Lipschitz continuity of harmonic maps. The following propositionis a consequence of the result by Korevaar and Schoen [10, Th. 2.4.6].
Proposition 4.
Under the hypotheses above, any r -equivariant harmonic map u is Lip-schitz continuous and its Lipschitz constant is bounded above by C · E / ( u ) , where theconstant C depends on the manifold M and its metric only. Now let G be a finite connected graph without terminals and G be its fundamentalgroup. By T we denote the universal covering tree of G . Similarly to the discussion above,for a locally rectifiable r -equivariant map u : T → Y the length density function l ( u ) is G -invariant and, hence, descends to the quotient G = T / G . In particular, by the length L ( u ) we8nderstand the integral R G l ( u ) dt . It is straightforward to see that if a map u minimises thelength among all locally rectifiable r -equivariant maps, then its restriction to every edge isa geodesic. If the latter has constant-speed parameterisation on every edge, then it is alsoharmonic and the length of every edge u I satisfies the relation L ( u ) = E ( u I )( b − a ) , see [4,Lemm. 12.5]. Conversely, if u is a r -equivariant harmonic map, then its restriction to everyedge is a constant-speed geodesic whose squared length is proportional to the energy asabove. In particular, the length is constant on the set of r -equivariant harmonic maps,where it achieves its minimum.
3. Proofs of the width inequalities
We start with the following lemma.
Main Lemma I.
Let M be a compact Riemannian manifold without boundary and ( Y , d ) be a locally compact Hadamard space. Let r : G → Isom ( Y ) be a representation of the fun-damental group G = p ( M ) . Suppose that the moduli space Harm, formed by r -equivariantharmonic maps, is non-empty and bounded in L -metric. Then there exists a positive con-stant C ⋆ with the following property: for any r -equivariant locally W , -smooth map uthere exists a harmonic map ¯ u ∈ Harm such thatd ( u , ¯ u ) C ⋆ ( E / ( u ) − E / ⋆ ) , (3.1) where E ⋆ = E ( ¯ u ) is the energy minimum among r -equivariant maps.Proof. First, note that inequality (3.1) is invariant under the rescaling of the metric on thetarget space Y . Hence, it is sufficient to prove the lemma under the assumption thatthe distance d ( u , ¯ u ) is not less than one. (3.2)Suppose the contrary. Then there exists a sequence of maps u k such that for any ¯ u ∈ Harmd ( u k , ¯ u ) > k ( E / ( u k ) − E / ( ¯ u )) . For each u k choose a harmonic map ¯ u k at which the infimum d ( u k , ¯ u k ) = inf { d ( u , v ) : v ∈ Harm } is attained. Such a harmonic map clearly exists: it is the value of u k under the nearest pointprojection onto Harm . (The lower semicontinuity of the energy [10, Th. 1.6.1] and Prop. 3imply that
Harm is a closed convex subset in the Hadamard space of r -equivariant locally L -integrable maps.)Denote by H ks , where s ∈ [ , ] , a geodesic homotopy between ¯ u k and u k ; we set H k = ¯ u k and H k = u k . Assuming that the parameter s is proportional to the arc length, we obtain d ( H ks , H k ) = s · d ( u k , ¯ u k ) > s · k ( E / ( u k ) − E / ( ¯ u k )) . Recall the energy E / ( · ) is convex along geodesic homotopies; s ( E / ( u k ) − E / ( ¯ u k )) > E / ( H ks ) − E / ( H k ) . Combining the last two inequalities we conclude that d ( H ks , H k ) > k ( E / ( H ks ) − E / ( H k )) . (3.3)9ow choose a sequence of s k ∈ [ , ] such that the distance d ( H ks k , H k ) equals one; bythe assumption (3.2) this is possible. Then relation (3.3) implies that the sequence E ( H ks k ) converges to E ⋆ as k → + ¥ . Since the moduli space Harm is bounded in L -metric, thelatter together with the choice of the s k ’s implies that the sequence H ks k is bounded in the W , -sense; that is d ( H ks k , w ) + E ( H ks k ) C , (3.4)where w is a fixed r -equivariant map. Now by the version of Rellich’s embedding theo-rem [10, Th. 1.13] we can find a subsequence H ks k (denoted by the same symbol) whichconverges in L -metric and point-wise to a locally W , -smooth map ¯ v . By the lower semi-continuity of the energy [10, Th. 1.6.1] the map ¯ v is energy minimising and by the point-wise convergence is r -equivariant. By the choice of the s k ’s we clearly have d ( H k , H ks k ) = d ( H k , H k ) − d ( H ks k , H k ) = d ( u k , ¯ u k ) − . Thus, the L -distance between the maps u k and v can be estimated as d ( u k , ¯ v ) d ( H k , H ks k ) + d ( H ks k , ¯ v ) = d ( u k , ¯ u k ) + ( d ( H ks k , ¯ v ) − ) . For sufficiently large k the second term on the right-hand side is negative and we arrive ata contradiction with the choice of the harmonic maps ¯ u k ’s.The following lemma summarises known results (essentially due to [10]) on the modulispace Harm , formed by r -equivariant maps. Lemma 1.
Let M be a compact Riemannian manifold without boundary and Y be a locallycompact Hadamard space. Let G be the fundamental group of M and r : G → Isom ( Y ) beits representation whose image does not fix a point on the ideal boundary of Y . Then themoduli space Harm, formed by r -equivariant harmonic maps, is non-empty and compactin C -topology. Since there is no direct reference for the statement on the compactness of
Harm and tomake our paper more self-contained, we give a proof now.
Proof of Lemma 1.
First, we explain the existence of a r -equivariant harmonic map.By [10, Th. 2.6.4] there exists an energy minimising sequence { u i } of equivariant Lip-schitz continuous maps, whose Lipschitz constants are uniformly bounded. Let W be afundamental domain for the action of G on the universal cover ˜ M . We claim that under thehypotheses of the theorem the ranges u i ( W ) are contained in a bounded subset of Y . In-deed, suppose the contrary. Then there exists a point x ∈ W such that the sequence { u i ( x ) } is unbounded in Y , i.e. d ( u i ( x ) , Q ) → + ¥ for some Q ∈ Y . For any g ∈ G consider the sequence d ( r ( g ) · u i ( x ) , u i ( x )) . By the equivariance of the u i ’sand the uniform boundedness of their Lipschitz constants we have d ( r ( g ) · u i ( x ) , u i ( x )) Cd ( g · x , x ) , and hence the quantities on the left hand side remain bounded as i → + ¥ . By the convexityof the distance between geodesics, relation (2.3), we see that the (Hausdorff) distancesbetween the geodesic segments Qu i ( x ) and r ( g ) · Qu i ( x ) also remain bounded as i → + ¥ .Since Y is locally compact, we can find a subsequence of u i , denoted by the same symbol,10uch that the segments Qu i ( x ) converge on compact subsets to a geodesic ray s with initialpoint at Q . Then the distance between s and r ( g ) · s is also bounded for any g ∈ G . Thisshows that s represents a fixed point for the action of r ( G ) and leads to a contradiction.Now, since Y is locally compact, the Arzela-Ascoli theorem applies and we can finda subsequence of u i converging in C -topology to an energy-minimising and, hence, har-monic map. Thus, the moduli space Harm is non-empty.Finally, we explain the compactness of
Harm . Let u i be a sequence of r -equivariantharmonic maps. By Prop. 3 their energies coincide and Prop. 4 the u i ’s are uniformly Lip-schitz continuous. The same argument as above shows that the ranges u i ( W ) are containedin a bounded subset of Y . Again by the Arzela-Ascoli theorem there exists a convergingsubsequence. By the lower semi-continuity of the energy the limit map is energy minimis-ing and, hence, harmonic. Thus, the moduli space Harm is compact in C -topology among r -equivariant harmonic maps. Proof of Theorem 1.
By Lemma 1 Main Lemma I applies: for given r -equivariant maps u and v we can find harmonic r -equivariant maps ¯ u and ¯ v such that d ( u , ¯ u ) and d ( v , ¯ v ) are estimated as in (3.1). By Lemma 1 the moduli space Harm is compact and, hence, thedistance between d ( ¯ u , ¯ v ) is uniformly bounded. The L -width of a geodesic homotopy H between u and v is the distance d ( u , v ) , and by the triangle inequality we have W ( H ) d ( u , ¯ u ) + d ( ¯ u , ¯ v ) + d ( v , ¯ v ) . The second term is bounded, and the first and the last can be estimated as in (3.1); thus, weobtain W ( H ) C ⋆ ( E / ( u ) + E / ( v )) + C , where C equals diam ( Harm ) − C ⋆ E / ⋆ . Since, under the hypotheses of the theorem, theenergy minimum E ⋆ is positive, this inequality can be re-written in the form (1.3).Now we explain the proof of Theorem 2; it follows essentially the same idea. First, wediscuss the version of Main Lemma I. By d ¥ ( u , v ) we denote below the maximum of thedistance function between maps u and v . Main Lemma II.
Let G be a finite graph and ( Y , d ) be a locally compact Hadamard space.Let r : G → Isom ( Y ) be a representation of the fundamental group G = p ( G ) . Supposethat the moduli space Harm, formed by r -equivariant harmonic maps T → Y , is non-emptyand compact in C -topology. Then there exists a positive constant C ⋆ with the followingproperty: for any continuous rectifiable r -equivariant map u there exists a harmonic map ¯ u ∈ Harm such that d ¥ ( u , ¯ u ) C ⋆ ( L ( u ) − L ⋆ ) , (3.5) where L ⋆ = L ( ¯ u ) is the length minimum among r -equivariant maps.Proof. First, without loss of generality we may assume that the maps u : T → Y underconsideration are such that their restrictions to every edge are parameterised proportionallyto the arc-length. Second, as in the proof of Main Lemma I, it is sufficient to prove thelemma under the assumption that the distance d ¥ ( u , ¯ u ) is not less than one.Suppose the contrary. Then there exists a sequence of maps u k and harmonic maps ¯ u k such that d ¥ ( u k , ¯ u k ) > k ( L ( u k ) − L ⋆ ) ;we suppose that the ¯ u k ’s minimise the distance { d ¥ ( u k , ¯ u ) , where u ∈ Harm } . Denote by H ks , where s ∈ [ , ] , a geodesic homotopy between ¯ u k and u k . Assuming that the parameter11s proportional to the arc-length and using the convexity of the length, relation (2.5), weobtain d ¥ ( H ks ) > k ( L ( H ks ) − L ( H k )) . Choosing a sequence s k ∈ [ , ] such that the left-hand side above equals to one, we con-clude that L ( H ks k ) converges to L ⋆ as k → + ¥ . Since the lengths of H ks k are bounded andthe edges of the H ks k ’s are parameterised proportionally to the arc-length, we see that thesequence of the H ks k ’s is equicontinuous. Further, the compactness of Harm implies thatthe latter sequence is d ¥ -bounded. Now the Arzela-Ascoli theorem applies and there existsa subsequence converging in d ¥ -metric to a continuous map ¯ v . The map ¯ v is clearly r -equivariant and length-minimising. Moreover, it has a constant-speed parametrisation and,hence, is harmonic. Now one gets a contradiction in the same way as in the proof of MainLemma I. Proof of Theorem 2.
First, Lemma 1 carries over the case of r -equivariant maps of trees.In more detail, we need to start with a length minimising sequence which is uniformlyLipschitz continuous. The latter can be constructed by re-parameterising any length min-imising sequence proportionally to the arc-length on every edge. The rest of the proof (ofLemma 1) carries over without essential changes.Now we simply follow the lines in the proof of Theorem 1 and use Main Lemma IIinstead of Main Lemma I.We proceed with the proofs of Theorems 3 and 4. First, recall some notation. Let L be a group acting properly and co-compactly by isometries on Y . For a homomorphism r : G → L by Z we denote the centraliser of the image r ( G ) in L . The group Z acts naturallyon the space of r -equivariant maps u : ˜ M → Y and, in particular, on the moduli space Harm . Lemma 2.
Under the hypotheses of Theorem 3, the moduli space Harm, formed by r -equivariant harmonic maps, is non-empty and the quotient Harm / Z is compact in C -topology.Proof. We start with the existence of a r -equivariant harmonic map. By [10, Th. 2.6.4.]there exists an energy minimising sequence { u i } of equivariant Lipschitz continuous maps,whose Lipschitz constants are uniformly bounded. Let W and D be fundamental domainsfor the actions of G on ˜ M and L on Y respectively. Fix a point x ∗ ∈ W . Then there exists asequence of elements h i ∈ L such that the maps h i · u i send x ∗ into the closure of D . Since the h i ’s are isometries, the sequence { h i · u i } is also energy minimising and uniformly Lipschitzcontinuous. Moreover, since L acts co-compactly, its fundamental domain D is bounded,and the uniform Lipschitz continuity implies that the ranges h i · u i ( W ) are contained ina bounded subset of Y . By the Arzela-Ascoli theorem there exists a subsequence, alsodenoted by h i · u i , converging to a limit map v .Now we define a homomorphism j : G → L such that the limit map v is j -equivariant.For this fix a generator g ∈ G and consider the points v ( g · x ) = lim ( h i · u i )( g · x ) and v ( x ) = lim ( h i · u i )( x ) , where x ∈ W . The triangle inequality implies that ( h i r ( g ) h − i ) · v ( x ) → v ( g · x ) as i → + ¥ . Now, since the action of L is proper, the sequence h i r ( g ) h − i contains a constant subse-quence; we denote it value by j ( g ) ∈ L . We use the h i ’s of this subsequence for the same12rocedure for another generator in G . Repeating the process we define j on all generators.It then extends as a homomorphism j : G → L and the map v is j -equivariant. As a resultof this procedure, we also have a sequence h i ∈ L such that h i r ( g ) h − i = j ( g ) for any g ∈ G . This identity implies that the h i ’s can be written in the form k · ¯ h i , where ¯ h i ∈ Z , and theelement k ∈ L conjugates r and j . Now, since the sequence h i · u i converges to v , thesequence ¯ h i · u i converges to k − v . Moreover, the latter is energy minimising and is formedby r -equivariant maps. Thus, the limit map k − v is a harmonic r -equivariant map and theexistence is demonstrated.The compactness of Harm / Z follows by the same argument as above with the substi-tution of the sequence of harmonic maps for the energy minimising sequence { u i } . ByProp. 3 the former sequence is also energy minimising, and by Prop. 4 is uniformly Lips-chitz continuous; the argument above yields a sequence ¯ h i ∈ Z such that ¯ h i · u i converges toa r -equivariant harmonic map. Proof of Theorem 3.
Let H be a fundamental domain for the action of Z on the mod-uli space Harm . First, Main Lemma I holds under a weaker hypothesis than the L -boundedness of Harm . More precisely, it is sufficient to assume that the domain H isbounded in the L -metric. Indeed, since the group Z acts by isometries, one can supposethat the maps ¯ u k ’s (in the proof of Main Lemma I) belong to H . The boundedness of thelatter is then used to obtain the W , -boundedness of the sequence H ks k , relation (3.4). Therest of the proof stays unchanged.Now the combination of Lemma 2 and estimate (3.1) yields the statement in the fashionsimilar to the proof of Theorem 1. Proof of Theorem 4.
First, Main Lemma II holds under a weaker hypothesis than the com-pactness of the moduli space
Harm . Similarly to the above, it is sufficient to assume thata fundamental domain for the action of Z on Harm is compact. Further, Lemma 2 carriesover the case of r -equivariant maps of trees; the proof follows essentially the same lineof argument. The combination of this version of Lemma 2 with estimate (3.5) yields thestatement in the same fashion as above.
4. Finitely generated subgroups in isometry groups
Recall that the action of a group L on a metric space ( Y , d ) by isometries defines an orbitpseudo-metric on L : d y ( g , h ) = d ( g · y , h · y ) , where g , h ∈ L , and y ∈ Y is a fixed reference point. For another point ¯ y ∈ Y the pseudo-metrics d y and d ¯ y are coarsely isometric; that is there exists a constant C ( = d ( y , ¯ y ) ) such that d ¯ y ( g , h ) − C d y ( g , h ) d ¯ y ( g , h ) + C . First, we show that the L ¥ -width inequalities imply an estimate for the conjugating elementin the orbit pseudo-metric. Lemma 3.
Let G be a semi-simple Lie group all of whose simple factors are non-compact.Let L be an irreducible lattice in G and ( a i ) i N be a finite list of elements in L whichdoes not fix a point on the ideal boundary of the associated symmetric space. Then for any onjugate (in L ) list ( b i ) i N any conjugating element g ∈ L , b i = g − a i g, satisfies theinequality d y ( g , e ) C ⋆ N (cid:229) i = ( d y ( a i , e ) + d y ( b i , e )) , where y ∈ Y is a reference point, and the constant depends only on the conjugacy class ofthe list ( a i ) .Proof. Let Y be a symmetric space associated with the Lie group G . Under the hypotheseson G , the natural G -invariant Riemannian metric on Y defines a distance d which makes Y into an Hadamard space.Consider the bouqet of N copies of a circle; denote by G = ⊕ Ni = Z its fundamentalgroup and by T its universal cover. Define a homomorphism r : G → L by the rule: thegenerator of the i th copy of Z maps into a i . For a fixed reference point y ∈ Y consider thegraph in Y whose vertices are points g · y , where g is a word in the alphabet ( a i ) . The edgesare geodesic arcs; two points g · y and g · y are joined by an edge if and only if g − g isan element a i or its inverse. Suppose that each edge is parameterised proportionally to thearc-length. Such a parametrisation defines a r -equivariant map u : T → Y , whose length L ( u ) is given by the sum (cid:229) Ni = d ( a i y , y ) .Analogously, for a conjugate list ( b i ) i N one defines a ( g − r g ) -equivariant map v : T → Y , where g is a conjugating element. Note that the map g · v is r -equivariant andits length L ( g · v ) coincides with L ( v ) = (cid:229) Ni = d ( b i y , y ) . By the hypotheses of the lemma,Theorem 2 applies and we have d ( g · y , y ) W ¥ ( H ) C ⋆ ( L ( u ) + L ( v )) , where H is a homotopy between u and g · v . Now the combination with the expressions forthe lengths finishes the proof. Proof of Theorem 5.
The statement is a direct consequence of Lemma 3 and the solutionof Kazhdan’s conjecture in [11]. The latter says that the word metric (with respect tosome finite set of generators) on an irreducible lattice L is quasi-isometric to the orbitmetric (with respect to the action on the associated symmetric space or Euclidean building)provided G is semi-simple and its rank > Lemma 4.
Let Y be a locally compact Hadamard space and L be a group acting properlyand co-compactly by isometries on Y . Then for any finite conjugate lists ( a i ) i N and ( b i ) i N of elements in L there exists an element g ∈ L with b i = g − a i g such thatd y ( g , e ) C ⋆ N (cid:229) i = ( d y ( a i , e ) + d y ( b i , e )) + C , where y ∈ Y is a reference point, and the constants depend only on the conjugacy class ofthe list ( a i ) .Proof. The proof follows the same line of argument as the proof of Lemma 3 with the useof Theorem 4 instead of Theorem 2.
Proof of Theorem 6.
By ˇSvarc-Milnor lemma [1, I.8.19] the word and orbit metrics on L are quasi-isometric. The combination of this with Lemma 4 implies the first statement ofthe theorem. Further, by [1, III. G .1.4] the word problem in L is soluble. This yields thealgorithm deciding the conjugacy of finite lists in the following fashion. If there existsan element conjugating two given lists, then it belongs to the finite subset of L formed byelements satisfying the bound (1.5). Using the solubility of the word problem, the algorithmchecks all elements from this finite set. 14 eferences [1] Bridson, M., Haefliger, A. Metric spaces of non-positive curvature.
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