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Unitary representations of fundamental groups and the spectrum of twisted Laplacians
UNITARY
REPRESENTATIONS
OF FUNDAMENTAL
129
GROUPS
53. THE INEQUALITY C&p, 1)’ j A,,(p)
We fix a set A of generators universal
covering
manifold
of x1(M), and let 9 c fi be a fundamental
fi. Given
a smooth
section
domain
in the
s of E,, we find
Thus we have
We shall prove
LEMMA 2. and s.
Proof. joining
s%
I(s(ol)-s(2)((2dx
I C
s2
We first note that s(aZ) - s(Z) =
1 and a~?. Thus, putting
Ildsl\2d-fx or some constant C not depending on c
ds, where E: [0, l] --t fi is a smooth path in 6l sP
c(t) = 63(?(t)), we have
s s 1
Ils(cJ5z)-s(~)(12 I
0
1
IIdsk(t) II2
0
II44 II2dt.
Without loss of generality, we may assume that M is orientable and CJis represented by a simple closed curve co in M. For each simple curve c:S’ -+ M, there exists an embedding F,:S’xD”-‘~MwithF(S’xO=c,wheren=dimM,andD”-’istheclosedunitdiscin IX”-‘. We put T, = F,(S’ x D”- ‘), which is a tubular neighborhood of the curve c. For each point x in M, we can find a curve cr: [0, l] + M without self-intersection such that c,(O) = x, {c,(l)} = c,(S’) nc,([O,
there exist finitely lift
of
satisfying
the closed
simple closed curve c, with c,(O) = x. Since M is compact
by a smooth
a
11). We then approximate
many
ci satisfying
curve c~.c~.c;~ and M = u TX, xeM
c1 = c,, , . . , cN = cxN such that M = fi Ti (K = TJ. Let Ei be i=l c”,(l) = oci(0), and let Fi: II3x D”-’ ---f2 be a lift of F,,
~il[O, l] x 0 = Ci. Put
z = F,([O, l] x D”-l).
We may assume
that
9 c
6
c
i=l
since M = 6
T, and find
i=l
s 54
Ils(o.%-s(5Z)))2d.2 I
2
i=l
SC2 i=l
s F,
1)s(a52)- s(2) I]2dZ
ssT,
1 IIds(c,W II’ dt dx, 0
where we have put c,(t) = F(s + t, y) (F = Fci, and x = F(s, y)). We write dx =.f(s, y) dsd”- ly,
130
Toshikazu Sunada
where ds and d”-‘y II d~(c,W))
are the Lebesgue densities on S’ and D”-‘. Thus II2 dtdx
= s
D”-’ ssSL S’
I (maxf) = (maxf)
IIW’(s
+
s Ljn-1sss1 ,$I s D”-I s SI
t,
Y) II2 dtf(s,
y) dsdy”- I,
lJds(F(s+t))(12dtdsdy”-’
ll~~~~(~,y)/12~~~y”-1
Ilds(F(s,~)ll~f(~,y)dsdy”-~ = C
s TX
IldSl12dx,
so that
s 9
Ils(oZ) - s(n) I(2d2 < C
IIdsl12dx
This completes the proof. Remark. In case of finite dimensional representations, the estimate C&p, Q)2I A,(p) can be obtained by using the argument in S. Gallot [S]. In fact, for s E Cm(Ep), we have n
~AMlldsII* = (A,ds,
ds) - IIVds(12-
2
(ds(Riccie,), ds(e,)),
k=l
where (ek} is an orthonormal
basis of T’M (ds = d,s). In particular, if APs = Ao(p)s, then
+A,IIdsl12 < Clldsll* where C = sup,?,(p) + max I(Riccie, e)l, ecTM P llell = 1
so that AM/Ids/lI C’lldsjl for some constant depending only on M. According to [6], we have sup IIdsll* I C”
s
lldsl12,
from which the desired inequality follows easily.
$4. WEAK CONTAINMENT
AND SPECTRUM
Let G be a discrete group with a finite set A of generators, and p and p1 unitary representations of G on Hilbert spaces V and Vi respectively. We say that p is weakly contained in p1 and write p < pl, if for every positive E and every orthonormal set
UNITARY
{Vl, . . . >c’,f1 in
REPRESENTATIONS
OF FUNDAMENTAL
ICci3
P(a)vj>V-
131
set {ul, . . . , u,} in VI such that
V there exists an orthonormal
(*)
GROUPS
C”i3
Pl(a)Uj)VII
<
&3
where ~JE A. Remark
1. If p is a sub-representation
of pr, then p < pl.
Remurk 2. II < p if and only if S(Q, p) = 0. Indeed, IIp(4r - v/I2 = 2 -(
Spect(A,)
p and p1 he unitary
this comes from the equality:
v> + (P(cJ -‘)v, v>). representations
of nl(M).
If p < pl,
then
c Spect(A,,).
It is known that if G is amenable, then every representation is weakly contained infinite direct sum of the regular representation of G (see [13]), hence we have
in an
COROLLARY. Let 61: X -+ M be a normal covering with covering transformation group G. If G is amenable, then Spect(A,) c Spect(A,) for every unitary representation p of G. In particular, Spect(A,) c Spect(A,).
Proof qf Theorem 2. We retain the terminology in $2. Let 1”~ Spect(A,). We fix a complete orthonormal basis {ri >$ 1 of V. Let c:be an arbitrary small positive number. From the definition of spectrum, we may take s E CS(Ep) such that (lA,s - is 11i2 I E and I(sllL2= 1. We shall construct a section t E C”(E,,) such that llb,, t - l.tl/i, I CE and litljLz 2 C’ > 0 for constitute some constants C and C’ not depending on E. Since {s(x)} wEg and {@;ds(x)},,~ compact sets in V, applying the BanachhSteinhaus theorem, we can select an integer n such that, for all vector v of the form s(x) or dq;ds(x). 2 u-
i;
/I
/I v -
Take an orthonormal the constant section
We then find, on x E 6,
we have
I
Vi)Vi
&
i=l 2
t
(v,
(TVi)cmi
i=l
II I
E.
set {ui}l= 1 in VI satisfying (*). For each vi (resp. ui), let sia (resp. ti,) be of E, (resp. Epl) on L?, defined in the same way as in $2, and put t(x)
Obviously
(v,
=
1 a
cPatx)C
(s(x), I
sia(x)>tia(x).
132
Toshikazu Sunada
Using the equality
II$(
v, cwi)cmi -
i
(v, Vi)Ui
i=l
I2=IIT’ +
2
V,
II
are not greater
-C
Vi)Vi
I
/I
i,j
)/
and
Ilda.(F((&
Sia)tia-(b
Sib)tib))li
(i
than 2sup lA(pB1’&+ 2/)s(12&and 2sup I(&, )(‘E + 2Ildsll’~ respectively.
here that (jds(l& I (IAs(jL2 (Is(IL2I &l/2 + I, so that not depending on E. On the other hand, we have
JILlI
(V,
2Rex(- (au,, Uj)X
ACHEC((S,~i,)ti,-(S,Sib)tib)
we find that
~Vi)(TVi
=
jl
C
*
II
2
Note
((At - Atlj& I CE for some constant
11
~~~~
Si6)fi6//i
‘11
~~,S,((S,Sia)tib-(S,Sib)tib)
’ II
hence ijtll$>
+ - (+ + 3 vol(M))e 2 a (for sufficiently
Acknowledgements
-The
author
would like to acknowledge
small E). This completes
helpful conversations
with A. Kastuda
the proof.
and T. Adachi.
REFERENCES I. M. F. ATIYAH: Elliptic operators, Discrete groups and von Neumann algebra, A&risque 32-33 (1976), 43-72. 2. R. BROOKS: The fundamental group and the spectrum of the Laplacian, Comment. Math. Helvetici 56 (1981), 581-598. 3. R. BROOKS: Combinatorial problems in spectral geometry, in the Proceeding of the Taniguchi Symposium Curvature and topology of Riemannian manijolds 1985, Springer Lect. Note 1201, 14-32. 4. R. BROOKS: The spectral geometry of tower of coverings, J. 013 Geom. 23 (1986), 97-107. 5. M. BURGER: Estimation de petites valeurs propres du Laplacien d’un revetement de varittts Riemanniennes compactes, C. R. Acad. Sci. Paris 302 (1986), 191-194. 6. J. CHEEGER: A lower bound for the smallest eigenvalue of the Laplacian, in Gunning, Problem in Analysis, (1970), 195-199. 7. J. M. G. FELL: Weak containment and induced representations of groups, Can. J. Math. 14 (1962), 237-268. 8. S. GALLOT: A Sobolev inequality and some geometric applications, in Spectra ofRiemannian manifolds, Kaigai Publ. Tokyo, (1983), 45-55. 9. F. P. GREENLEAF: Invariant Means on Topological groups and Their Application, van Nostrand. Reinhold, (1969). 10. A. KAIXUDA and T. SLJNADA:Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. (To appear). 11. D. A. KAZHDAN: Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1987), 63-65. 12. T. SUNADA: Number theoretic analogues in spectral geometry, in Proc. DD6. Shanghai, Springer Lect. Note. 13. R. J. ZIMMER: Ergodic Theory and Semisimple Groups, Birkhiuser, Boston (1984).
Department
of Mathematics
Nagoya
University
Nagoya
464
Japan
REPRESENTATIONS
OF FUNDAMENTAL
129
GROUPS
53. THE INEQUALITY C&p, 1)’ j A,,(p)
We fix a set A of generators universal
covering
manifold
of x1(M), and let 9 c fi be a fundamental
fi. Given
a smooth
section
domain
in the
s of E,, we find
Thus we have
We shall prove
LEMMA 2. and s.
Proof. joining
s%
I(s(ol)-s(2)((2dx
I C
s2
We first note that s(aZ) - s(Z) =
1 and a~?. Thus, putting
Ildsl\2d-fx or some constant C not depending on c
ds, where E: [0, l] --t fi is a smooth path in 6l sP
c(t) = 63(?(t)), we have
s s 1
Ils(cJ5z)-s(~)(12 I
0
1
IIdsk(t) II2
0
II44 II2dt.
Without loss of generality, we may assume that M is orientable and CJis represented by a simple closed curve co in M. For each simple curve c:S’ -+ M, there exists an embedding F,:S’xD”-‘~MwithF(S’xO=c,wheren=dimM,andD”-’istheclosedunitdiscin IX”-‘. We put T, = F,(S’ x D”- ‘), which is a tubular neighborhood of the curve c. For each point x in M, we can find a curve cr: [0, l] + M without self-intersection such that c,(O) = x, {c,(l)} = c,(S’) nc,([O,
there exist finitely lift
of
satisfying
the closed
simple closed curve c, with c,(O) = x. Since M is compact
by a smooth
a
11). We then approximate
many
ci satisfying
curve c~.c~.c;~ and M = u TX, xeM
c1 = c,, , . . , cN = cxN such that M = fi Ti (K = TJ. Let Ei be i=l c”,(l) = oci(0), and let Fi: II3x D”-’ ---f2 be a lift of F,,
~il[O, l] x 0 = Ci. Put
z = F,([O, l] x D”-l).
We may assume
that
9 c
6
c
i=l
since M = 6
T, and find
i=l
s 54
Ils(o.%-s(5Z)))2d.2 I
2
i=l
SC2 i=l
s F,
1)s(a52)- s(2) I]2dZ
ssT,
1 IIds(c,W II’ dt dx, 0
where we have put c,(t) = F(s + t, y) (F = Fci, and x = F(s, y)). We write dx =.f(s, y) dsd”- ly,
130
Toshikazu Sunada
where ds and d”-‘y II d~(c,W))
are the Lebesgue densities on S’ and D”-‘. Thus II2 dtdx
= s
D”-’ ssSL S’
I (maxf) = (maxf)
IIW’(s
+
s Ljn-1sss1 ,$I s D”-I s SI
t,
Y) II2 dtf(s,
y) dsdy”- I,
lJds(F(s+t))(12dtdsdy”-’
ll~~~~(~,y)/12~~~y”-1
Ilds(F(s,~)ll~f(~,y)dsdy”-~ = C
s TX
IldSl12dx,
so that
s 9
Ils(oZ) - s(n) I(2d2 < C
IIdsl12dx
This completes the proof. Remark. In case of finite dimensional representations, the estimate C&p, Q)2I A,(p) can be obtained by using the argument in S. Gallot [S]. In fact, for s E Cm(Ep), we have n
~AMlldsII* = (A,ds,
ds) - IIVds(12-
2
(ds(Riccie,), ds(e,)),
k=l
where (ek} is an orthonormal
basis of T’M (ds = d,s). In particular, if APs = Ao(p)s, then
+A,IIdsl12 < Clldsll* where C = sup,?,(p) + max I(Riccie, e)l, ecTM P llell = 1
so that AM/Ids/lI C’lldsjl for some constant depending only on M. According to [6], we have sup IIdsll* I C”
s
lldsl12,
from which the desired inequality follows easily.
$4. WEAK CONTAINMENT
AND SPECTRUM
Let G be a discrete group with a finite set A of generators, and p and p1 unitary representations of G on Hilbert spaces V and Vi respectively. We say that p is weakly contained in p1 and write p < pl, if for every positive E and every orthonormal set
UNITARY
{Vl, . . . >c’,f1 in
REPRESENTATIONS
OF FUNDAMENTAL
ICci3
P(a)vj>V-
131
set {ul, . . . , u,} in VI such that
V there exists an orthonormal
(*)
GROUPS
C”i3
Pl(a)Uj)VII
<
&3
where ~JE A. Remark
1. If p is a sub-representation
of pr, then p < pl.
Remurk 2. II < p if and only if S(Q, p) = 0. Indeed, IIp(4r - v/I2 = 2 -(
Spect(A,)
p and p1 he unitary
this comes from the equality:
v> + (P(cJ -‘)v, v>). representations
of nl(M).
If p < pl,
then
c Spect(A,,).
It is known that if G is amenable, then every representation is weakly contained infinite direct sum of the regular representation of G (see [13]), hence we have
in an
COROLLARY. Let 61: X -+ M be a normal covering with covering transformation group G. If G is amenable, then Spect(A,) c Spect(A,) for every unitary representation p of G. In particular, Spect(A,) c Spect(A,).
Proof qf Theorem 2. We retain the terminology in $2. Let 1”~ Spect(A,). We fix a complete orthonormal basis {ri >$ 1 of V. Let c:be an arbitrary small positive number. From the definition of spectrum, we may take s E CS(Ep) such that (lA,s - is 11i2 I E and I(sllL2= 1. We shall construct a section t E C”(E,,) such that llb,, t - l.tl/i, I CE and litljLz 2 C’ > 0 for constitute some constants C and C’ not depending on E. Since {s(x)} wEg and {@;ds(x)},,~ compact sets in V, applying the BanachhSteinhaus theorem, we can select an integer n such that, for all vector v of the form s(x) or dq;ds(x). 2 u-
i;
/I
/I v -
Take an orthonormal the constant section
We then find, on x E 6,
we have
I
Vi)Vi
&
i=l 2
t
(v,
(TVi)cmi
i=l
II I
E.
set {ui}l= 1 in VI satisfying (*). For each vi (resp. ui), let sia (resp. ti,) be of E, (resp. Epl) on L?, defined in the same way as in $2, and put t(x)
Obviously
(v,
=
1 a
cPatx)C
(s(x), I
sia(x)>tia(x).
132
Toshikazu Sunada
Using the equality
II$(
v, cwi)cmi -
i
(v, Vi)Ui
i=l
I2=IIT’ +
2
V,
II
are not greater
-C
Vi)Vi
I
/I
i,j
)/
and
Ilda.(F((&
Sia)tia-(b
Sib)tib))li
(i
than 2sup lA(pB1’&+ 2/)s(12&and 2sup I(&, )(‘E + 2Ildsll’~ respectively.
here that (jds(l& I (IAs(jL2 (Is(IL2I &l/2 + I, so that not depending on E. On the other hand, we have
JILlI
(V,
2Rex
ACHEC((S,~i,)ti,-(S,Sib)tib)
we find that
~Vi)(TVi
=
jl
C
*
II
2
Note
((At - Atlj& I CE for some constant
11
~~~~
Si6)fi6//i
‘11
~~,S,((S,Sia)tib-(S,Sib)tib)
’ II
hence ijtll$>
+ - (+ + 3 vol(M))e 2 a (for sufficiently
Acknowledgements
-The
author
would like to acknowledge
small E). This completes
helpful conversations
with A. Kastuda
the proof.
and T. Adachi.
REFERENCES I. M. F. ATIYAH: Elliptic operators, Discrete groups and von Neumann algebra, A&risque 32-33 (1976), 43-72. 2. R. BROOKS: The fundamental group and the spectrum of the Laplacian, Comment. Math. Helvetici 56 (1981), 581-598. 3. R. BROOKS: Combinatorial problems in spectral geometry, in the Proceeding of the Taniguchi Symposium Curvature and topology of Riemannian manijolds 1985, Springer Lect. Note 1201, 14-32. 4. R. BROOKS: The spectral geometry of tower of coverings, J. 013 Geom. 23 (1986), 97-107. 5. M. BURGER: Estimation de petites valeurs propres du Laplacien d’un revetement de varittts Riemanniennes compactes, C. R. Acad. Sci. Paris 302 (1986), 191-194. 6. J. CHEEGER: A lower bound for the smallest eigenvalue of the Laplacian, in Gunning, Problem in Analysis, (1970), 195-199. 7. J. M. G. FELL: Weak containment and induced representations of groups, Can. J. Math. 14 (1962), 237-268. 8. S. GALLOT: A Sobolev inequality and some geometric applications, in Spectra ofRiemannian manifolds, Kaigai Publ. Tokyo, (1983), 45-55. 9. F. P. GREENLEAF: Invariant Means on Topological groups and Their Application, van Nostrand. Reinhold, (1969). 10. A. KAIXUDA and T. SLJNADA:Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. (To appear). 11. D. A. KAZHDAN: Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1987), 63-65. 12. T. SUNADA: Number theoretic analogues in spectral geometry, in Proc. DD6. Shanghai, Springer Lect. Note. 13. R. J. ZIMMER: Ergodic Theory and Semisimple Groups, Birkhiuser, Boston (1984).
Department
of Mathematics
Nagoya
University
Nagoya
464
Japan