TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 4, April 2011, Pages 1743–1763 S 0002-9947(2010)05238-9 Article electronically published on November 5, 2010
THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY FOR BUSEMANN NONPOSITIVE CURVATURE SPACES
MAMORU TANAKA
Abstract. For an isometric action of a finitely generated group on the ultra- limit of a sequence of global Busemann nonpositive curvature spaces, we state a sufficient condition for the existence of a fixed point of the action in terms of the energy of equivariant maps from the group into the space. Further- more, we show that this energy condition holds for every isometric action of a finitely generated group on any global Busemann nonpositive curvature space in a family which is stable under ultralimit, whenever each of these actions has afixedpoint. We also discuss the existence of a fixed point of affine isometric actions of a finitely generated group on a uniformly convex, uniformly smooth Banach space in terms of the energy of equivariant maps.
1. Introduction One of the purposes of this paper is to generalize results in [6] and [7] for Hadamard spaces to global Busemann nonpositive curvature spaces: For a family of global Busemann nonpositive curvature spaces which is stable under ultralimit, we investigate whether any isometric action of a finitely generated group on any space in the family has a fixed point, in terms of the energy of equivariant maps from the group into spaces in the family. Let Γ be a finitely generated group and ρ a homomorphism from Γ into the full isometry group of a global Busemann nonpositive curvature space (Definition 4.1). The homomorphism ρ can be regarded as an isometric action on the space. For the homomorphism ρ, we define a generalized nonnegative energy functional E (Definition 3.1) on the space of the ρ-equivariant maps from Γ into the space. The energy functional vanishes at a ρ-equivariant map if and only if the image of the mapisafixedpointofρ(Γ). In [7], the gradient flow of the energy of equivariant maps into Hadamard spaces is used to investigate the existence of a fixed point. The gradient flow was intro- duced by Jost in [9] and Mayer in [11]. It generates a strongly continuous semigroup of Lipschitz-continuous mappings and decreases the energy in the most efficient way.
Received by the editors May 16, 2008. 2010 Mathematics Subject Classification. Primary 58E20, 58E40; Secondary 51F99. Key words and phrases. Global Busemann NPC spaces, fixed point, ultralimits, uniformly convex, uniformly smooth Banach space. The author was supported by Grant-in-Aid for JSPS Fellows (21·1062).
c 2010 American Mathematical Society Reverts to public domain 28 years from publication 1743
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1744 MAMORU TANAKA
Although it seems that it is difficult to generalize Jost-Mayer’s gradient flow to global Busemann nonpositive curvature spaces, we can consider the absolute gra- dient |∇−E|(f) of the energy functional E at each ρ-equivariant map f (Definition 5.1). The absolute gradient gives the maximum descent of the energy functional around each map with respect to a natural distance on the space of ρ-equivariant maps. The absolute gradient |∇−E|(f) can be regarded as the norm of the gradient of E at f. If the absolute gradient vanishes at a ρ-equivariant map, then the map minimizes the energy functional. We can always find a sequence of ρ-equivariant maps such that the absolute gradient approaches 0 (Lemma 5.4). For a nonprincipal ultrafilter on N (Definition 2.1), the ultralimit of a sequence of global Busemann nonpositive curvature spaces with base points is always defined (Definition 2.2). The ultralimit is also defined for a sequence of homomorphisms from Γ into the full isometry groups of the spaces, equivariant maps, and energies under a certain condition. Theorem 1. Let ρ be a homomorphism from Γ into the full isometry group of a global Busemann nonpositive curvature space (N,d).DenotebyE the energy functional on the space of ρ-equivariant maps. Take a sequence of ρ-equivariant maps fn such that the absolute gradient of E approaches 0. If there exists a positive constant C such that 2 |∇−E|(fn) ≥ CE(fn) holds for every n, then, for any nonprincipal ultrafilter on N,thereexistsafixed point of the ultralimit of the sequence of homomorphisms ρn ≡ ρ in the ultralimit of the sequence of spaces (Nn,dn) ≡ (N,d) with base points fn(e).Heree denotes the identity element of Γ. In the case of a Hadamard space, in [7], they prove that if the assumptions in Theorem 1 hold, then there exists a fixed point of the image of the original homomorphism ρ in the original space. Let L be a family of global Busemann nonpositive curvature spaces which is stable under ultralimit (Definition 2.3). Theorem 2. Suppose that, for any (N,d) ∈Land homomorphism ρ from Γ into the full isometry group of (N,d), there exists a fixed point of ρ(Γ). Then there exists a positive constant C such that 2 |∇−Eρ|(f) ≥ CEρ(f) holds for all (N,d) ∈L, homomorphisms ρ from Γ into the full isometry group of (N,d),andρ-equivariant maps f from Γ into N.HereEρ denotes the energy functional for ρ-equivariant maps. The constant C = C(Γ, L) should be independent of (N,d), ρ,andf. Examples of a family of global Busemann nonpositive curvature spaces which is stable under ultralimit are given in Section 6. In [6], they prove Theorem 2 in the case of a family of Hadamard spaces which is stable under ultralimit. It is well known that if L is the family of all Hilbert spaces, then the so-called Kazhdan’s property (T ) is equivalent to the existence of a constant in Theorem 2. Hence the maximum of the constant C(Γ, L) in Theorem 2 can be regarded as a generalization of the Kazhdan constant of Γ. Another purpose of this paper is to consider the existence of a fixed point of affine isometric actions of a finitely generated group Γ on a Banach space (B,)interms
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1745
of the energy of equivariant maps. Let π be a linear isometric Γ-representation on B, that is, a homomorphism from Γ into the full linear isometry group of (B,). Denote by Bπ(Γ) the closed subspace of fixed vectors of π.TheΓ- representation π descends to a linear isometric Γ-representation π on the quotient space B/Bπ(Γ).Letρ be an affine isometric Γ-representation on B,thatis,ρ is written as ρ(γ)v = ρ0(γ)v +o(γ) by a linear isometric Γ-representation ρ0 on B and amapo from Γ into B. Note that an isometry on a real Banach space is affine; see [3]. The Γ-representation ρ also descends to an affine isometric Γ-representation ρ ρ0(Γ) on B/B .DenotebyEρ the energy functional on the space of all ρ -equivariant maps.
Definition ([1]). (i) We say that Γ has property (FB) if any affine isometric Γ- representation on B has a fixed point. (ii) We say that Γ has property (TB) if, for any nontrivial linear isometric Γ- representation π on B, there exists a positive constant C(π) such that max v − π (γ)v≥C(π) γ∈S for all v ∈ B/Bπ(Γ) satisfying v =1,whereS is a finite generating subset of Γ. In [1], they pointed out that, according to a theorem by Guichardet, if Γ has property (FB), then it has property (TB). In this paper, we introduce a new prop- erty defined in terms of the energy of equivariant maps and investigate a relation between property (FB) and the new one.
Definition. We say that Γ has property (EB) if there exists a positive constant C(ρ) for any affine isometric Γ-representation ρ on B such that 2 |∇−Eρ |(f) ≥ C(ρ)Eρ (f) for all ρ -equivariant maps f.
Theorem 3. If Γ has property (FB), then it has property (EB).
There exists a finitely generated Abelian group which has property (TB) but does not have property (FB) for some Banach space B; see [1, Example 2.22]. However, as is well known, a finitely generated group has property (TH ) for all Hilbert spaces H (i.e., has Kazhdan’s property (T )) if and only if it has property (FH ) for all Hilbert spaces H. Furthermore, these are also equivalent to having property (EH ) for all Hilbert spaces H. Thus one may expect that there is some relation between property (TB) and property (EB) for a Banach space B.Butthere is a finitely generated group which has property (TB) but does not have property (EB) for some Banach space B, and the author does not know whether a finitely generated group which has property (EB) has property (TB). However we can show Proposition 4 below, which gives a relation between property (TB) and a property which is somewhat weaker than property (EB). Definition. We say that Γ has property (EB) if, for any affine isometric Γ- representation ρ on B whose linear part ρ0 is nontrivial, there exists a positive constant C(ρ) such that 2 |∇−Eρ |(f) ≥ C(ρ)Eρ (f) for all ρ -equivariant maps f.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1746 MAMORU TANAKA
Proposition 4. Let Γ be a finitely generated Abelian group, and B a uniformly convex, uniformly smooth Banach space. If Γ has property (TB), then it has property (EB) . From Proposition 4, we can show that there exists a finitely generated Abelian group which has property (EB) but does not have property (FB)forsomeBanach space B. The paper is organized as follows: We review the definitions and properties of the ultralimit of a sequence of metric spaces in Section 2, the energy of equivariant maps in Section 3, global Busemann nonpositive curvature spaces in Section 4, and the absolute gradient in Section 5. We show Theorem 1 and Theorem 2 in Section 6. In Section 7, we review the definitions of the properties (FB), (TB), (EB)and (EB) , and show Theorem 3 and Proposition 4. The author would like to express his gratitude to Professor Hiroyasu Izeki for suggesting this topic and helpful comments.
2. Ultralimit of metric spaces In this section, we recall the definition and some properties of the ultralimit of a sequence of metric spaces. Basic references of this section are [2], [10], and [6]. Definition 2.1. A nonprincipal ultrafilter ω on N is a subset of 2N satisfying the following: (i) ∅ ∈ ω and N ∈ ω; (ii) if A ∈ ω and A ⊂ B,thenB ∈ ω; (iii) if A ∈ ω and B ∈ ω,thenA ∩ B ∈ ω; (iv) A ∈ ω or N\A ∈ ω for any subset A ⊂ N;(v) B ∈ ω for any finite subset B ⊂ N. An example of a nonprincipal ultrafilter on N can be given as follows: The set ω := {B ⊂ N : |N \ B| < ∞} satisfies (i), (ii), (iii), and (v). Considering all ω ⊂ 2N satisfying (i), (ii), (iii), (v), and ω ⊂ ω, we obtain a maximal one, ω0,byZorn’s Lemma. If there exists A ⊂ N such that A ∈ ω0 and N \ A ∈ ω0, then the set ω0 ∪{C ⊂ N : A ∩ B ⊂ C for some B ∈ ω0} also satisfies (i), (ii), (iii), and (v). The set ω0 is contained properly in this set. This contradicts that ω0 is maximal. Hence ω0 is a nonprincipal ultrafilter. Let ω be a nonprincipal ultrafilter on N. For any bounded sequence of real numbers an there exists a unique real number l such that {n : |an − l| <}∈ω for any >0. Denote l by aω or ω-limn an. If a sequence an converges to a real number a in the usual sense, then aω = a. Thus we can regard aω as a limit of an in some sense. Moreover, ω-lim is linear, and if a pair of bounded sequences an,bn satisfies an ≤ bn for every n,thenaω ≤ bω holds.
Definition 2.2. Let ω be a nonprincipal ultrafilter on N and (Mn,dn) a sequence of metric spaces with base points on ∈ Mn.DenotebyM∞ the set of sequences (pn) such that pn ∈ Mn and dn(pn,on) is bounded independently of n.Wesaythat (pn)and(qn) ∈ M∞ are equivalent if ω-limn dn(pn,qn)=0.Letpω or ω-limn pn denote the equivalence class of (pn), and Mω denote the set of the equivalence classes. Endow Mω with a metric dω(pω,qω):=ω-limn dn(pn,qn). We also write (Mω,dω)=ω-limn(Mn,dn,on), and call it the ultralimit of a sequence of metric spaces (Mn,dn) with base points on with respect to ω. The ultralimit of a sequence of metric spaces is complete. Note that if a Banach space (B,) is finite dimensional, then ω-limn(B,,o)isisometricto(B,). But if (B,) is infinite dimensional, then ω-limn(B,,o)maynotbeisometric to (B,).
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1747
Definition 2.3. A family L of metric spaces is said to be stable under ultralimit if for any sequence of metric spaces (Nn,dn)inL with base points on and any sequence of positive real numbers rn, there exists a nonprincipal ultrafilter ω on N such that ω-limn(Nn,rndn,on)isinL.
3. Energy functionals and harmonic maps In this section, we define the space of equivariant maps from a finitely generated group into a metric space, the energy functional on the space. A basic reference of this section is [6]. Let Γ be a finitely generated group, S the finite generating subset of Γ which −1 −1 satisfies S = S := {s : s ∈ S} and does not contain the identity element e of Γ. → −1 Let m : S (0, 1) be a function satisfying γ∈S m(γ)=1andm(γ)=m(γ )for every γ ∈ S. The function m is called a weight on S.Anisometry on a metric space (M,d)isamapg : M → M which is surjective and satisfies d(g(x),g(y)) = d(x, y) for all x, y ∈ M.DenotebyIsom(M,d) the group which consists of all isometries on M. For a homomorphism ρ :Γ→ Isom(M,d), M denotes the space of ρ-equivariant maps from Γ into M,whereρ-equivariant means that f(γα)=ρ(γ)f(α) for every γ,α ∈ Γ. Endow M with a metric dM(f,g)=dM(f(e),g(e)). Then we can identify (M,dM)with(M,d) through the correspondence f → f(e). Definition 3.1. We define an energy functional E : M→[0, ∞)by 1 E(f):= m(γ)d(f(e),f(γ))2 2 γ∈S for f ∈M.
The energy functional is continuous on (M,dM). A ρ-equivariant map f is called harmonic if f minimizes E.NotethatE(f) = 0 if and only if f(e) is a fixed point of ρ(Γ). Let ω be a nonprincipal ultrafilter on N.Let(Mn,dn) be a sequence of metric spaces with base points on,andρn :Γ→ Isom(Mn,dn) a sequence of homomor- phisms. It is easy to see that the following Lemma 3.2 and Lemma 3.3 hold.
Lemma 3.2. Suppose that, for every γ ∈ Γ, dn(on,ρn(γ)on) is bounded indepen- dently of n. Then we can define a homomorphism ρω :Γ→ Isom(Mω,dω) by ρω(γ)pω := ω-limn(ρn(γ)pn) for every γ ∈ Γ and pω ∈ Mω,where(pn) ∈ M∞ is a representative of pω.
Denote by Mn the set of all ρn-equivariant maps from Γ into Mn,andbyMω the set of all ρω-equivariant maps from Γ into Mω.DenotebyM∞ the set of all sequences fn ∈Mn such that dn(on,fn(e)) is bounded independently of n.
Lemma 3.3. Suppose that, for every γ ∈ Γ, dn(on,ρn(γ)on) is bounded indepen- dently of n.For(fn) ∈M∞, the limit map fω :Γ→ Mω defined by fω(γ):=ω- limn fn(γ) is ρω-equivariant. Conversely, for any f ∈Mω, there exists a represen- tative (fn) ∈M∞ of f.
Denote by En the energy functional on Mn.
Lemma 3.4. Suppose that fω ∈Mω has a representative (fn) ∈M∞ for which En(fn) is bounded independently of n. Then the energy functional Eω : Mω →
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1748 MAMORU TANAKA
[0, ∞) defined by 1 E (f )= m(γ)d (f (e),f (γ))2 ω ω 2 ω ω ω γ∈S ∈M satisfies Eω(fω)=ω-limn En(fn) for any representative (fn) ∞ of fω.
Proof. By assumption, dn(fn(e),fn(γ)) is bounded independently of n and γ ∈ S. Hence there exists C > 0 such that dn(fn(e),fn(γ))
for all f,g ∈Mn. Hence we obtain |E (f ) − E (f )| = |E (f )1/2 − E (f )1/2|(E (f )1/2 + E (f )1/2) n n n n √ n n n n n n√ n n 1/2 ≤ 2dM (f ,f )(2E (f ) + 2dM (f ,f )) √ n n n n n n n n < 2( 2C1/2 + ) for every n ∈ K .SinceK ∈ ω,wehave √ { ∈ N | − | 1/2 }∈ n : En(fn) En(fn) < 2( 2C + ) ω for any >0. This implies that ω-limn En(fn)=ω-limn En(fn).
4. Global Busemann nonpositive curvature spaces In this section, we recall the definition and some properties of global Busemann nonpositive curvature spaces. A global Busemann nonpositive curvature space was introduced by Busemann in [4]. Most of what appears in this section is included in [8] and [13]. Let (M,d) be a metric space. For a pair of points p, q ∈ M,ashortest geodesic joining p to q is an isometric embedding c of a closed interval [0,l]intoM such that c(0) = p and c(l)=q.Notethatl = d(p, q). A metric space is called a geodesic space if any pair of points has a shortest geodesic joining them.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1749
Definition 4.1. A complete geodesic space (N,d) is called a global Busemann nonpositive curvature space (global Busemann NPC space)if
l l 1 d(c ( 1 ),c ( 2 )) ≤ d(c (l ),c (l )) 1 2 2 2 2 1 1 2 2
holds for all pairs of shortest geodesics ci :[0,li] → N (i =1, 2) satisfying c1(0) = c2(0). This inequality is called the Busemann NPC inequality.
Example 4.2. Hadamard spaces are global Busemann NPC spaces. Complete sim- ply connected Riemannian manifolds with nonpositive sectional curvature, trees, Euclidean Bruhat-Tits buildings, and Hilbert spaces are Hadamard spaces. A strictly convex Banach space (i.e., a Banach space whose unit sphere does not have a nontrivial line segment) is also a global Busemann NPC space. In particu- lar, the Lebesgue space Lp([0, 1]) such that p = 2 is not Hadamard space, but, for 1 The convexity of the distance function in the following theorem is the most important feature of global Busemann NPC spaces. Theorem 4.3 (see [8]). Let c0 :[0,l0] → N, c1 :[0,l1] → N be shortest geodesics in a global Busemann NPC space (N,d).Thend(c0(l0t),c1(l1t)) is a convex function on [0, 1]. From this theorem, any pair of points in a global Busemann NPC space can be joined by precisely one shortest geodesic. Hence global Busemann NPC spaces are simply connected. Furthermore, this theorem implies that the energy functional on a global Busemann NPC space is convex along a shortest geodesic. The ultralimit of a sequence of geodesic spaces is a geodesic space. But the ultralimit of a sequence of global Busemann NPC spaces may not be a global 2 Busemann NPC space. For example, Banach spaces (R , p)with1 1 1 n 2 2 n 1/n u − vn =(|u − v | + |u − v | ) =1, 1 1 n 2 2 n 1/n 1/n u − wn =(|u − w | + |u − w | ) =2 , 1 1 n 2 2 n 1/n n 1/n v − wn =(|v − w | + |v − w | ) =(2 +1) , v + w v1 + w1 v2 + w2 1 u − =(|u1 − |n + |u2 − |n)1/n = . 2 n 2 2 2 ∈ R2 ≡ ≡ ≡ Thus, for (un), (vn)and(wn) n∈N with un u, vn v,andwn w,we have uω − vωω =1,uω − wωω =1,andvω − wωω = 2. On the other hand, uω =( vω + wω)/2 holds. Since the line segment from u to w is a shortest geodesic, the line segment from vω to wω is also a shortest geodesic. Since uω −vωω +uω − wωω = vω − wωω, the path from vω through uω to wω whose part between vω and uω is a line segment and the part between uω and wω is also a line segment is 2 another shortest geodesic. Hence in the ultralimit ω-limn(R , n, 0) the shortest License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1750 MAMORU TANAKA geodesic between two points is not unique. Therefore the ultralimit is not a global Busemann NPC space. 5. Absolute gradient In this section, we define the absolute gradient and investigate its properties. Let (M,dM) be a complete geodesic space, and F : M→[0, ∞) a lower semicontinuous convex function. Here ‘convex’ means convex along all shortest geodesics. Definition 5.1. We define the absolute gradient |∇−F | of F at f ∈Mby F (f) − F (g) |∇−F |(f):=max lim sup , 0 . g→f,g∈M dM(f,g) For Hadamard spaces, the following Proposition 5.2 and Proposition 5.3 are proved by Mayer in [11]. The proofs are valid for complete geodesic spaces. Proposition 5.2. For f ∈M, we have F (f) − F (g) |∇−F |(f)=max sup , 0 . g= f,g∈M dM(f,g) Proposition 5.3. Suppose |∇−F | is bounded. Then we have the following: (i) |∇−F | is lower semicontinuous. (ii) Apointf minimizes F if and only if |∇−F |(f)=0holds. Note that, as we saw in the proof of Lemma 3.4, the energy functional E satisfies √ 1/2 |E(f) − E(g)|≤2dM(f,g)( 2E(f) + dM(f,g)). This implies | − | √ E(f) E(g) 1/2 |∇−E|(f) ≤ lim sup ≤ 2 2E(f) g→f,g∈M dM(f,g) for all f ∈M. In particular, the absolute gradient |∇−E| is bounded. In [11], Mayer constructed a gradient flow of a lower semicontinuous convex function F on a Hadamard space. Although, on global Busemann NPC spaces, it seems that it is difficult to construct such a gradient flow by Mayer’s method, we can prove the following: Lemma 5.4. inf |∇−F |(f)=0. f∈M Proof. Suppose that C := inff∈M |∇−F |(f)ispositive.Then F (f) − F (g) C ≤ sup g= f, g∈M dM(f,g) holds for all f ∈Mby Proposition 5.2. In particular, F (f) > 0 for all f ∈M. Take f0 ∈Mand 0 <<1 arbitrarily and set F (f0) − F (g) A0 := g ∈M\{f0} :(1− )C ≤ . dM(f0,g) By the definition of C, A0 is not empty. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1751 ∈ ≤ If infg∈A0 F (g) =0,takef1 A0 such that F (f1) (1 + )infg∈A0 F (g)andset F (f1) − F (g) A1 := g ∈M\{f1} :(1− )C ≤ . dM(f1,g) By the definition of C, A1 is not empty. Since f1 ∈ A0, for any g ∈ A1,wehave − − − F (f0) F (g) ≥ (F (f0) F (f1)) + (F (f1) F (g)) dM(f0,g) dM(f0,f1)+dM(f1,g) − − ≥ (1 )CdM(f0,f1)+(1 )CdM(f1,g) dM(f0,f1)+dM(f1,g) =(1− )C. ∈ ⊂ Hence g A0 holds, that is, A1 A0.Thusinfg∈A1 F (g) = 0 holds. Inductively, ∈ N ∈ ≤ i for each i ,wecantakefi Ai−1 such that F (fi) (1 + )infg∈Ai−1 F (g), and set F (fi) − F (g) Ai := g ∈M\{fi} :(1− )C ≤ = ∅. dM(fi,g) ⊂ ∈ N ∈ Then we have Ai Ai−1 for each i and infg∈Ai F (g) =0.Thusforg Ai we have F (fi) − F (g) dM(f ,g) ≤ i (1 − )C i (1 + )inf ∈ F (g) − inf ∈ F (g) ≤ g Ai−1 g Ai (1 − )C i (1 + )inf ∈ F (g) − inf ∈ F (g) ≤ g Ai−1 g Ai−1 (1 − )C i inf ∈ F (g) = g Ai−1 . (1 − )C Since fi ∈ Ai−1 and F (fi) ≤ F (fi−1) hold for each i ∈ N,wehave i i F (fi) F (f0) dM(f ,g) ≤ ≤ i (1 − )C (1 − )C for all g ∈ Ai. Therefore, for any > 0, there exists i ∈ N such that i F (f0) dM(f ,f ) ≤ diam A ≤ 2 < j k i (1 − )C ≥ M ∈M for every j, k i.Since is complete, fi converges to some f∞ .Thus ∞ ∪{ } { } i=0(Ai fi )iseitheranemptysetoraone-pointset f∞ . On the other hand, the function 1/dM(fi,g) is continuous on the open set M\{fi},and−F (g) is upper semicontinuous on M.Thus F (fi) − F (g) Fi (g):= dM(fi,g) M\{ } { ∈M\{ } } is upper semicontinuous on fi . Hence g fi : Fi (g) = M\(Ai ∪{fi}) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1752 MAMORU TANAKA ∪{ } ∈ ∪{ } is open, that is, Ai fi is closed. Hence f∞ Ai fi for every i. This implies ∞ ∪{ } { } ∈M\{ } that i=0(Ai fi )= f∞ . However, by assumption, there exists g0 f∞ such that F (f∞) − F (g ) (1 − )C ≤ 0 . dM(f∞,g0) Since f∞ ∈ Ai ∪{fi} and Ai+1 ∪{fi+1}⊂Ai,wehavef∞ ∈ Ai for each i. Hence F (g0) ∈ ∞ ∪{ } { } for every i. This implies that g0 i=1(Ai fi )= f∞ ,thatis,f∞ = g0.This contradicts g0 ∈M\{f∞}. ∈ ≤ If infg∈A0 F (g)=0,takef1 A0 such that F (f1) F (f0), and set − ∈M\{ } − ≤ F (f1) F (g) A1 := g f1 :(1 )C . dM(f1,g) By the definition of C, A1 is not empty and is a subset of A0.Ifinfg∈A F (g) =0, 1 then we can deduce a contradiction as in the case of infg∈A0 F (g) =0.Otherwise ∈ ≤ we take f2 A1 such that F (f2) F (f1), and set − ∈M\{ } − ≤ F (f2) F (g) A2 := g f2 :(1 )C . dM(f2,g) Inductively, for each i ∈ N,ifinf ∈ F (g) = 0, then we can deduce a contradic- g Ai−1 ∈ tion as in the case of infg∈A0 F (g) = 0. Otherwise we take fi Ai−1 such that ≤ F (fi ) F (fi−1), and set − ∈M\{ } − ≤ F (fi ) F (g) ∅ Ai := g fi :(1 )C = . dM(fi ,g) ⊂ ∈ N ∈ Then we have Ai Ai−1 for each i .Thusforg Ai we have − i F (fi ) F (g) F (fi−1) F (f0) dM(f ,g) ≤ ≤ ≤ . i (1 − )C (1 − )C (1 − )C ∈M ∞ ∪{ } { } Hence fi converges to some f∞ and i=0(Ai fi )= f∞ . Therefore we can deduce a contradiction as in the case of f∞. Corollary 5.5. There exists a sequence in M such that the absolute gradient ap- proaches 0. If the sequence converges to a point, then the point minimizes F . Proof. By Lemma 5.4, the existence of such a sequence fn ∈Mis obvious. By Proposition 5.3, |∇−F | is lower semicontinuous. Hence we have |∇−F |(f∞) ≤ lim |∇−F |(fn)=0. n→∞ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1753 6. The fixed-point property for a global Busemann NPC space In this section, we prove Theorem 1 and Theorem 2. Let Γ be a finitely generated group, S a finite generating subset, and m a weight as in Section 3. Proposition 6.1. Let ω be a nonprincipal ultrafilter, (Nn,dn) a sequence of global Busemann NPC spaces, and ρn :Γ→ Isom(Nn,dn) a sequence of homomorphisms. Denote by En the energy functional on the set of all ρn-equivariant maps from Γ into Nn.Takeaρn-equivariant map fn arbitrarily for each n,andsetfn(e) as a base point of (Nn,dn). Suppose that En(fn) is bounded independently of n and that |∇−En|(fn) → 0 as n →∞. Then there exists a limit map ω-limn fn,whichisa harmonic ρω-equivariant map. Proof. Since En(fn) is uniformly bounded independently of n,thereexistsC>0 such that dn(fn(e),fn(γ)) En(fn) − En(g) En(fn) − En(gn) |∇−En|(fn) ≥ sup ≥ . g= fn,g∈Mn dMn (fn,g) dMn (fn,gn) By the assumption that |∇−En|(fn) → 0asn →∞,weget − ≤ |∇ | ≤ ˆ|∇ | → En(fn) En(gn) dMn (fn,gn) −En (fn) C −En (fn) 0 as n →∞. By Lemma 3.4, we have Eω(fω) − Eω(g)=ω- lim(En(fn) − En(gn)) ≤ Cωˆ - lim |∇−En|(fn)=0. n n Therefore, we obtain Eω(fω) ≤ Eω(g) for all g ∈Mω,thatis,fω is harmonic. Using this proposition, we deduce the following two main theorems. Theorem 1. Let (N,d) be a global Busemann NPC space, ρ :Γ→ Isom(N,d) a homomorphism. Denote by E the energy functional on the space of ρ-equivariant maps. Take a sequence of ρ-equivariant maps fn satisfying |∇−E|(fn) → 0 as n →∞. If there exists a positive constant C such that 2 |∇−E|(fn) ≥ CE(fn) for every n, then, for any nonprincipal ultrafilter ω on N, there exists a fixed point of ρω in (Nω,dω):=ω-limn(N,d,fn(e)). Proof. Set (Nn,dn):=(N,d), and denote by En (= E) the energy functional for ρ-equivariant maps from Γ into Nn. By assumption, we have 2 CEn(fn) ≤|∇−En|(fn) → 0 as n →∞. Hence En(fn) → 0asn →∞. ThisimpliesthatEn(fn) is uniformly bounded independent of n. By Proposition 6.1, for any nonprincipal ultrafilter ω, there exists a ρω-equivariant harmonic map fω := ω-limn fn such that Eω(fω)=0. Therefore fω(e)isafixedpointofρω(Γ). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1754 MAMORU TANAKA Theorem 2. Let L be a family of global Busemann NPC spaces which is stable under ultralimit. Suppose that, for any (N,d) ∈Land ρ :Γ→ Isom(N,d),there exists a fixed point of ρ(Γ). Then there exists a positive constant C such that 2 |∇−EN,ρ|(f) ≥ CEN,ρ(f) holds for all (N,d) ∈L, homomorphisms ρ :Γ→ Isom(N,d) and ρ-equivariant maps f,whereEN,ρ denotes the energy functional for ρ-equivariant maps. The constant C = C(Γ, L) should be independent of (N,d), ρ,andf. Proof. The proof is by contradiction. Assume that the assertion is false. Then, for each n,thereexist(Nn,dn) ∈L, a homomorphism ρn :Γ→ Isom(Nn,dn)anda ρn-equivariant map fn :Γ→ Nn such that 2 1 |∇−E |(f ) < E (f ). Nn,ρn n n Nn,ρn n In particular, ENn,ρn (fn) > 0holdsforeveryn. Because of the assumption, there → exists a ρn-equivariant map gn :Γ Nn for each n which satisfies ENn,ρn (gn)=0. |∇ | Thus fn does not minimize ENn,ρn . By Proposition 5.3, we have −ENn,ρn (fn) > −1/2 0 for every n.Set(Nn,dn,fn(e)) := (Nn,ENn,ρn (fn) dn,fn(e)). We can regard → → ρn :Γ Isom(Nn,dn) as a homomorphism ρn :Γ Isom(Nn,dn). Moreover, we have ENn,ρn (fn) = 1 for every n; hence ENn,ρn (fn) is bounded independent of n. |∇ | 2 We have −EN ,ρn (fn) < 1/n for every n. Take a nonprincipal ultrafilter ω so ∈Ln that (Nω,dω) . By Proposition 6.1, the ρω-equivariant map fω := ω-limn fn minimizes ENω ,ρω . On the other hand, we have ENω,ρω (fω) = 1. This contradicts our assumption. We give some examples of a family of global Busemann NPC spaces which is stable under ultralimit. Example 6.2. For a Banach space (B,), the function δB():=inf{1 −u + v/2:u, v≤1 , u − v≥} is called the modulus of convexity of B.ABanachspaceB is said to be uniformly convex if δB() > 0 for any >0. Uniformly convex Banach spaces are strictly convex Banach spaces. For example, the Lebesgue space L ([0, 1]) satisfies p (p − 1)2/8+o(2)if1 {n ∈ N : | rnunn − 1| <η, | rnvnn − 1| <η,rnun − vnn >− η}∈ω. Henceweobtain r u r v r u − v − η n ∈ N : n n n < 1, n n n < 1, n n n n > ∈ ω. 1+η 1+η 1+η 1+η License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1755 This implies that r u + v − η n ∈ N : n n n n ≤ 1 − δ ∈ ω 2(1 + η) Bn 1+η for any 0 <η<. Therefore we have u + v − η ω ≤ inf (1 + η) 1 − δ =1− δ(), 2 η>0 1+η ≥ that is, δBω () δ() for any >0. Example 6.3. Fix k>0 arbitrarily. Let Lk be the family of global Busemann NPC spaces (N,rd), where r>0and(N,d) are global Busemann NPC spaces with the following condition: For any point p ∈ N and geodesic c :[0,l] → N, kd(p, c(tl))2 ≤ (1 − t)d(p, c(0))2 + td(p, c(l))2 − (1 − t)td(c(0),c(l))2 holds for all 0 ≤ t ≤ 1. Then the family Lk is stable under ultralimit. Note that the family L1 consists of all Hadamard spaces. The reason why Lk is stable under ultralimit is the following: For a non- principal ultrafilter ω and sequences (Nn,dn) ∈Lk and on ∈ Nn,set(Nω,dω)=ω- limn(Nn,dn,on). Take arbitrary p, q ∈ Nω,andlet(pn), (qn) ∈ N∞ be their repre- sentatives respectively. For a sequence of shortest geodesics cn :[0,ln] → Nn from pn to qn, the limit map cω :[0,lω] → Nω defined by cω(t):=ω-limn cn(tln/lω)isa shortest geodesic from p to q. Take a shortest geodesicc ˜ :[0,l] → Nω from p to q, and representations (˜cn(t)) ∈ N∞ ofc ˜(t)foreacht ∈ [0,l], where l = lω.Bythe definition of Lk,wehave 2 kdn(˜cn(tl),cn(tln)) 2 2 2 ≤ (1 − t)dn(˜cn(tl),cn(0)) + tdn(˜cn(tl),cn(ln)) − (1 − t)tdn(cn(0),cn(ln)) 2 2 =(1− t){dn(˜cn(tl),cn(0)) − (tdn(cn(0),cn(ln))) } 2 2 +t{dn(˜cn(tl),cn(ln)) − ((1 − t)dn(cn(0),cn(ln))) } 2 2 2 2 =(1− t){dn(˜cn(tl),pn) − (tln) } + t{dn(˜cn(tl),qn) − ((1 − t)ln) } for any 0 ≤ t ≤ 1. Hence we obtain 2 2 2 2 2 kdω(˜c(tl),cω(tlω)) ≤ (1 − t){(tl) − (tlω) } + t{((1 − t)l) − ((1 − t)lω) } =0 for all 0 ≤ t ≤ 1. Therefore the shortest geodesic from p to q is unique, and it is the ultralimit of shortest geodesics. Hence the Busemann NPC inequalities for (Nn,dn) imply the Busemann NPC inequality for (Nω,dω). By a similar argument, for any point p ∈ Nω and geodesic c :[0,l] → Nω,wehave 2 2 2 2 kdω(p, c(t)) ≤ (1 − t)dω(p, c(0)) + tdω(p, c(l)) − (1 − t)tdω(c(0),c(l)) for all 0 ≤ t ≤ l. Example 6.4. Fix k>0 arbitrarily. Let L k be a family of global Busemann NPC spaces (N,rd)wherer>0and(N,d) are global Busemann NPC spaces (N,d) with the following condition: For any point p ∈ N and geodesic c :[0,l] → N, d(p, c(t))2 ≤ (1 − t)d(p, c(0))2 + td(p, c(l))2 − k(1 − t)td(c(0),c(l))2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1756 MAMORU TANAKA holds for all 0 ≤ t ≤ 1. The metric space satisfying this inequality was introduced by Ohta in [12]. This inequality induces the inequality in the definition of Lk;see [12]. The family L k is also stable under ultralimit. The proof is the same as that for Lk. 7. The fixed-point property for Banach spaces In this section, we review the definitions of the properties (FB), (TB), (EB)and (EB) , and show Theorem 3 and Proposition 4. Let Γ be a finitely generated group, S a finite generating subset, and m aweight as in Section 3. Denote by O(B) the full linear isometry group of a Banach space (B,), and by Isom(B) the full affine isometry group of (B,). A linear isometric Γ-representation π on B is a homomorphism π :Γ→ O(B). Denote by Bπ(Γ) the closed subspace consisting of fixed vectors of π. We can define a linear isometric Γ-representation π on B/Bπ(Γ) by π(γ)[v]:=[π(γ)v]foreach[v] ∈ B/Bπ(Γ) and γ ∈ Γ, where [v] ∈ B/Bπ(Γ) means the equivalence class of v ∈ B. Similarly, an affine isometric Γ-representation ρ on B is a homomorphism ρ : Γ → Isom(B). An affine isometric Γ-representation ρ on B is written as ρ(γ)v = ρ0(γ)v + o(γ) by a linear isometric Γ-representation ρ0 on B and a map o :Γ→ B. We can define a linear isometric Γ-representation ρ on B := B/Bρ0(Γ) by ∈ ∈ ρ (γ)[v]:=[ρ(γ)v]=ρ0(γ)[v]+[o(γ)] for each [v] B and γ Γ. Definition 7.1. (i) We say that Γ has property (FB) if any affine isometric Γ- representation on B has a fixed point. (ii) We say that Γ has property (TB) if, for any nontrivial linear isometric Γ- representation ρ0, there exists a positive constant C(ρ0) such that − ≥ max [v] ρ0(γ)[v] C(ρ0) γ∈S for all [v] ∈ B/Bπ(Γ) satisfying [v] =1. (iii) We say that Γ has property (EB) if there exists a positive constant C(ρ)for any affine isometric Γ-representation ρ such that 2 |∇−Eρ |(f) ≥ C(ρ)Eρ (f) for all ρ -equivariant maps f. Then we have the following theorem. Theorem 3. If Γ has property (FB), then it has property (EB). Proof. Take an arbitrary affine isometric Γ-representation ρ :Γ→ Isom(B). Since Γ has property (FB), there exists a fixed point v0 ∈ B of ρ(Γ). We can regard afixedpoint[v0]ofρ (Γ) as the origin of B . We recall that if Γ has property (FB), then it has property (TB). Hence we have a positive constant C(ρ )such that maxγ∈S [v] − ρ (γ)[v]≥C(ρ ) for all [v] ∈ B satisfying [v] =1.If f(e) =0,thenEρ (f)=0≤|∇−Eρ |(f). Denote a linear action Id − ρ (γ) ∈ by T (γ)foreachγ S,whereId is the identity map on B .ThenEρ (f)= 2 − γ∈S m(γ) T (γ)f(e) /2. Set g(γ):=(1 )f(γ)for0<<1, which is a License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1757 ρ -equivariant map. If f(e) =0,thenwehave |∇−E |(f) ρ 2 2 ∈ m(γ)T (γ)f(e) − ∈ m(γ)T (γ)g(e) ≥ lim γ S γ S → 0 2f(e) − g(e) T (γ)f(e)−T (γ)g (e) T (γ)f(e) + T (γ)g (e) = m(γ) lim → 0 f(e) − g(e) 2 γ∈S T (γ)f(e) = m(γ) T (γ)f(e) f(e) γ∈S T (γ)f(e) ≥ min m(γ)max max T (γ)f(e) γ∈S γ∈S f(e) γ∈S ≥ min m(γ)C(ρ ) m(γ)T (γ)f(e). γ∈S γ∈S Using the H¨older inequality, we have 1/2 1/2 1 2 2 2 E (f) ≤ m(γ) T (γ)f(e) T (γ)f(e) ρ 2 γ∈S γ∈S 1 1/2 m(γ)2 1/2 ≤ m(γ)2T (γ)f(e)2 T (γ)f(e)2 ∈ 2 2 ∈ ∈ minγ S m(γ) γ S γ S m(γ)2T (γ)f(e)2 ≤ γ∈S 2minγ∈S m(γ) 2 ∈ m(γ)T (γ)f(e) ≤ γ S . 2minγ∈S m(γ) 2 2 3 Therefore we obtain |∇−Eρ |(f) ≥ 2C(ρ ) minγ∈S m(γ) Eρ (f). In order to state Proposition 4, we define a uniformly convex, uniformly smooth Banach space, which has good properties for investigating the absolute gradient. Definition 7.2. (i) A Banach space B is said to be uniformly convex if the modulus of convexity of B, δB():=inf{1 −u + v/2:u, v≤1 , and u − v≥}, is positive whenever >0. (ii) A Banach space B is said to be uniformly smooth if the modulus of smoothness of B, ∗ { − − ≤ ≤ } δB(τ):=sup u + v /2+ u v /2 1: u 1 , and v τ , ∗ → → satisfies δB(τ)/τ 0asτ 0. “Uniformly convex, uniformly smooth” is abbreviated to “ucus”. Ucus Banach spaces are global Busemann NPC spaces. Examples of ucus Banach spaces are Hilbert spaces and the Lebesgue spaces Lp([0, 1]) with 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1758 MAMORU TANAKA To prove Proposition 4, we prepare the following proposition. Proposition 7.3. Let B be a ucus Banach space and ρ :Γ→ Isom(B,) an affine isometric Γ-representation on B. Then we have ∗ T0(γ)v |∇−E|(f) ≥ m(γ)T (γ)f(e) Re(T (γ)f(e)) v γ∈S v =2 m(γ)T (γ)f(e) Re(T (γ)f(e))∗ v γ∈S for all ρ-equivariant maps f and nonzero vectors v ∈ B.HereT (γ):=Id − ρ(γ), − ∗ ∗ ∈ T0(γ):=Id ρ0(γ),andRe v (w) is the real part of v w for w B. The function ∗ γ∈S m(γ) T (γ)f(e) Re(T (γ)f(e)) is continuous on the space of ρ-equivariant maps. Proof. Take an arbitrary ρ-equivariant map f and a nonzero vector v ∈ B.Set g(γ):=ρ(γ)(f(e)+v)for>0andγ ∈ Γ. Computing as the proof of Theorem 3, we have T (γ)f(e) + T (γ)g(e) T (γ)f(e)−T (γ)g(e) |∇−E|(f) ≥ m(γ) 2 v γ∈S for any >0. We get ∗ T (γ)f(e)−T (γ)g(e)≥Re(T (γ)g(e)) (T (γ)f(e) − T (γ)g(e)) ∗ = Re(T (γ)g(e)) (T0(γ)v) for every γ ∈ S. On the other hand, since T (γ) is continuous, T (γ)g(e)con- verges to T (γ)f(e)as → 0. Because the duality map is uniformly continuous, ∗ ∗ (T (γ)g(e)) (T0(γ)v)convergesto(T (γ)f(e)) (T0(γ)v)as → 0 for every γ ∈ S. Therefore we have T (γ)f(e) + T (γ)g (e) T (γ)f(e)−T (γ)g (e) m(γ) lim →0 2 v γ∈S ≥ ∗ T0(γ)v m(γ) T (γ)f(e) Re(T (γ)f(e)) ∈ v γ S v v = m(γ)T (γ)f(e) Re(T (γ)f(e))∗ − Re(T (γ)f(e))∗ρ (γ) . v 0 v γ∈S ∗ ∗ Since ρ0(γ)isalinearisometry,wehave(T (γ)f(e)) ρ0(γ) = (T (γ)f(e)) =1 and ∗ −1 −1 (T (γ)f(e)) ρ0(γ) ρ0(γ )T (γ)f(e)=T (γ)f(e) = ρ0(γ )T (γ)f(e). ∗ −1 ∗ These equations imply (T (γ)f(e)) ρ0(γ)=(ρ0(γ )T (γ)f(e)) for every γ ∈ S. −1 −1 Since ρ0(γ )o(γ)+o(γ ) = 0 holds, we obtain −1 −1 −1 ρ0(γ )T (γ)f(e)=ρ0(γ )f(e) − ρ0(γ )ρ(γ)f(e) −1 −1 −1 = ρ0(γ )f(e) − ρ0(γ )ρ0(γ)f(e) − ρ0(γ )o(γ) −1 −1 = ρ0(γ )f(e) − f(e)+o(γ ) = −T (γ−1)f(e). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1759 ∗ −1 ∗ Therefore we get (T (γ)f(e)) ρ0(γ)=−(T (γ )f(e)) for every γ ∈ S.Using −1 −1 T (γ )f(e) = ρ0(γ )T (γ)f(e) = T (γ)f(e),wehave v v m(γ)T (γ)f(e) Re(T (γ)f(e))∗ − Re(T (γ)f(e))∗ρ (γ) v 0 v γ∈S v =2 m(γ)T (γ)f(e) Re(T (γ)f(e))∗ . v γ∈S Therefore we have ∗ v |∇−E|(f) ≥ 2 m(γ)T (γ)f(e) Re(T (γ)f(e)) . v γ∈S On the other hand, the functions T (γ)f(e) and Re(T (γ)f(e))∗ are continuous on the space of ρ-equivariant maps. Hence the function m(γ)T (γ)f(e) Re(T (γ)f(e))∗ γ∈S is also continuous on the space of ρ-equivariant maps. Next we review the definition of property (EB) and show Proposition 4. Definition 7.4. We say that Γ has property (EB) if, for any affine isometric Γ- representation ρ whose linear part ρ0 is nontrivial, there exists a positive constant C(ρ) such that 2 |∇−Eρ |(f) ≥ C(ρ)Eρ (f) for all ρ -equivariant maps f. Note that a nontrivial affine isometric action ρ whose linear part ρ0 is trivial does not have a fixed point on B, but Eρ (f) > 0=|∇−Eρ |(f)holdsforallρ -equivariant maps f. Hence, in Definition 7.4, we consider only affine isometric Γ-representations whose linear parts are nontrivial. Proposition 4. Let Γ be a finitely generated Abelian group, and B a ucus Banach space. If Γ has property (TB), then it has property (EB) . Proof. Take an arbitrary affine isometric Γ-representation ρ on B whose linear part ρ0 is nontrivial. The quotient space B is also a ucus Banach space; see [1]. By Proposition 7.3, we have ∗ Re(T (γ)f(e)) (T0(γ)v) |∇−E |(f) ≥ m(γ)T (γ)f(e) ρ v γ∈S for all ρ -equivariant maps f :Γ→B and nonzero vectors v ∈B .SetT (γ):=Id− − ρ (γ), T0(γ):=Id ρ0(γ), O := γ∈S m(γ)[o(γ)], X(f):= γ∈S m(γ)ρ0(γ)f(e), W (f):=X(f)+O,andV (f):=f(e) − W (f)=f(e) − X(f) − O. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1760 MAMORU TANAKA InthecaseofV (f) =0,wehave ∗ Re(T (γ)f(e)) (T0(γ)V (f)) ∗ =Re(T (γ)f(e)) (T0(γ)f(e) − T0(γ)W (f)) =Re(T (γ)f(e))∗(T (γ)f(e) − T (γ)W (f)) = T (γ)f(e)−Re(T (γ)f(e))∗T (γ)W (f) ≥T (γ)f(e)−{T (γ)f(e)+T (γ)W (f)−T (γ)f(e)} =2T (γ)f(e)−T (γ)f(e)+T (γ)W (f) ∈ for every γ S. Because κ∈S m(κ)=1,wehave T (γ)f(e)+T (γ)W (f) = m(κ)T (γ)f(e)+T (γ) m(κ)ρ (κ)f(e) ∈ ∈ κS κ S = m(κ)(T (γ)f(e)+T (γ)ρ (κ)f(e)) κ∈S for every γ ∈ S. Since Γ is Abelian, we have T (γ)ρ (κ)f(e)=ρ (κ)f(e) − ρ (γ)ρ (κ)f(e) = ρ (κ)f(e) − ρ (κ)ρ (γ)f(e) − = ρ0(κ)f(e) ρ0(κ)ρ (γ)f(e) = ρ0(κ)(T (γ)f(e)). Henceweget m(κ)(T (γ)f(e)+T (γ)ρ (κ)f(e)) = m(κ)((Id+ ρ0(κ))(T (γ)f(e)) ∈ ∈ κ S κ S ≤ m(κ) (Id + ρ0(κ))(T (γ)f(e)) κ∈S for every γ ∈ S. By property (TB), for γ ∈ S satisfying T (γ)f(e) =0,thereexists κγ ∈ S such that T (γ)f(e) T (γ)f(e) − ρ (κ ) ≥C(ρ ). T (γ)f(e) 0 γ T (γ)f(e) 0 Since B is uniformly convex, we have 1 T (γ)f(e) T (γ)f(e) 1 − + ρ (κ ) ≥δ (C(ρ )). 2 T (γ)f(e) 0 γ T (γ)f(e) B 0 Setting κγ = γ for γ ∈ S satisfying T (γ)f(e) =0,wehave − ≥ 2 T (γ)f(e) T (γ)f(e)+ρ0(κγ )T (γ)f(e) 2δB (C(ρ0)) T (γ)f(e) for every γ ∈ S. On the other hand, we have 1/2 V (f)≤ m(γ)T (γ)f(e)≤ m(γ)T (γ)f(e)2 . γ∈S γ∈S License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1761 Therefore we have Re(T (γ)f(e))∗(T (γ)V (f)) m(γ) 0 T (γ)f(e) V (f) γ∈S ∈ m(κ)(2T (γ)f(e)−(Id + ρ (κ))(T (γ)f(e))) ≥ m(γ) κ S 0 T (γ)f(e) V (f) γ∈S m(κ )(2T (γ)f(e)−(Id + ρ (κ ))(T (γ)f(e))) ≥ m(γ) γ 0 γ T (γ)f(e) V (f) γ∈S min ∈ m(κ)2δ (C(ρ ))T (γ)f(e) ≥ m(γ) κS B 0 T (γ)f(e) ( m(λ)T (λ)f(e)2)1/2 γ∈S λ∈S 1/2 2 =minm(κ)2δB (C(ρ0)) m(γ)T (γ)f(e) κ∈S ∈ √ γ S 1/2 =minm(κ)2 2δB (C(ρ0))Eρ (f) , κ∈S that is, 2 2 2 |∇−Eρ |(f) ≥ 8δB (C(ρ0)) min m(κ) Eρ (f). κ∈S 2 2 Set C(ρ):=8δB (C(ρ0)) minκ∈S m(κ) . InthecaseofV (f)=0andO =0,since f(e) − X(f)=O we have V (af)=af(e) − X(af) − O = a(f(e) − X(f)) − O =(a − 1)O =0 for a = 1. Thus, using the result in the case of V (f) = 0 and Proposition 7.3, we have 1/2 1/2 lim C(ρ) Eρ (af) a1 Re(T (γ)(af(e)))∗(T (γ)V (af)) ≤ lim m(γ) 0 T (γ)(af(e)) a1 V (af) γ∈S V (af) = lim 2 m(γ)T (γ)(af(e)) Re(T (γ)(af(e)))∗ a1 V (af) γ∈S −O =2 m(γ)T (γ)f(e) Re(T (γ)f(e))∗ O γ∈S ≤|∇−Eρ |(f). 2 Since Eρ is continuous, we have |∇−Eρ |(f) ≥ C(ρ)Eρ (f). InthecaseofV (f)=0andO = 0, we suppose f(e) = 0. By property (TB), ∈ − ≥ there exists γ1 S such that ρ0(γ1)f(e) f(e) C(ρ0) f(e) .SinceB is uniformly convex, we have ρ0(γ1)f(e)+f(e) 1 − >δ (C(ρ )). 2f(e) B 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1762 MAMORU TANAKA Since V (f)=0andO =0,wehavef(e)=X(f). Hence we have 2f(e) = f(e)+X(f) = m(γ)(f(e)+ρ0(γ)f(e)) ∈ γ S ≤ m(γ) f(e)+ρ0(γ)f(e) ∈ γ S − γ∈S,γ= γ1 ≤ 2f(e)−2δB ()m(γ1)f(e) < 2f(e). This is a contradiction. Hence f(e) = 0, and a ρ -equivariant map g with g(e) =0 satisfies V (ag) =0for0 ≤|∇−Eρ |(f). 2 Therefore we have |∇−Eρ |(f) ≥ C(ρ)Eρ (f). Note that the assumption that Γ is Abelian is only used in the case of V (f) =0. 2 Hence, if there exists a positive constant C satisfying |∇−Eρ |(f) ≥ CEρ (f)for all f ∈Mwith V (f) = 0, we can show Proposition 4 without the assumption. References 1. U. Bader, A. Furman, T. Gelander, and N. Monod, Property (T ) and rigidity for actions on Banach spaces,ActaMath.198 (2007), no.1, 57–105. MR2316269 (2008g:22007) 2. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. MR1744486 (2000k:53038) 3. Y. Benyamini and J. Lindestrauss, Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Provi- dence, RI, 2000. MR1727673 (2001b:46001) 4. H. Busemann, The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955. MR0075623 (17:779a) 5. A. Guichardet, Sur la cohomologie des groupes topologiques. II, Bull. Sci. Math. (2) 96 (1972), 305–332. MR0340464 (49:5219) 6. H. Izeki, T. Kondo and S. Nayatani, Fixed-point property of random groups, Ann. Global Anal. Geom. 35 (2009), no. 4, 363–379. MR2506240 7. H. Izeki and S. Nayatani, Combinatorial harmonic maps and discrete-group actions on Hadamard spaces, Geom. Dedicata 114 (2005), 147–188. MR2174098 (2006k:58024) 8. J. Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Math. ETH Z¨urich. Birkh¨auser Verlag, Basel, 1997. MR1451625 (98g:53070) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY 1763 9. J. Jost, Nonlinear Dirichlet forms, New directions in Dirichlet forms, 1–47, AMS/IP Stud. Adv. Math., 8, Amer. Math. Soc., Providence, RI, 1998. MR1652278 (99k:31012) 10. M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, 183. Birkh¨auser Boston, Inc., Boston, MA, 2001. MR1792613 (2002m:57018) 11. U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), 199–253. MR1651416 (99m:58067) 12. S. Ohta, Convexities of metric spaces, Geom. Dedicata 125 (2007), 225–250. MR2322550 (2008j:53063) 13. A. Papadopoulos, Metric spaces, convexity and nonpositive curvature, IRMA Lectures in Mathematics and Theoretical Physics, 6. European Mathematical Society (EMS), Z¨urich, 2005. MR2132506 (2005k:53042) Mathematical Institute, Tohoku University, Sendai 980-8578, Japan E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use