# Can time-dependent density functional theory predict the excitation energies
of conjugated polymers?

Jianmin Tao  Theoretical Division and CNLS, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545  Sergei Tretiak  Theoretical Division and CNLS,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545  Center for
Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New
Maxico 87545  Jian-Xin Zhu  Theoretical Division and CNLS, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545

###### Abstract

Excitation energies of light-emitting organic conjugated polymers have been
investigated with time-dependent density functional theory (TDDFT) within the
adiabatic approximation for the dynamical exchange-correlation potential. Our
calculations show that the accuracy of the calculated TDDFT excitation
energies largely depends upon the accuracy of the dihedral angle obtained by
the geometry optimization on ground-state DFT methods. We find that, when the
DFT torsional dihedral angles between two adjacent phenyl rings are close to
the experimental dihedral angles, the TDDFT excitation energies agree fairly
well with experimental values. Further study shows that, while hybrid density
functionals can correctly respect the thumb rule between singlet-singlet and
singlet-triplet excitation energies, semilocal functionals do not, suggesting
inadequacy of the semilocal functionals in predicting triplet excitation
energies of conjugated polymers.

###### pacs:

71.15.Mb, 31.15.ee, 71.45.Gm

The most important progress made in the development of molecular electronics
is the discovery of electroluminescent conjugated polymers thomas98  – that
is, fluorescent polymers that emit light when these polymers in the excited
states are stimulated by, say, electric current. Conjugated polymers are
organic semiconductors with delocalized $\pi$-molecular orbitals along the
polymeric chain. These materials are a major challenge to inorganic materials
which have been dominating the commercial market in light-emitting diodes for
display and other purpose pmay . The attraction of conjugated polymers lies at
their versatility, because their physical properties such as color purity and
emission efficiency can be fine-tuned by manipulation of their chemical
structures. The systematic modification of the properties of emissive polymers
by synthetic design has become a vital component in the optimization of light-
emitting devices.

Theoretical investigation of their optical absorption plays a significant role
in computer-aided design and optimization of the electroluminescent polymers.
The method of choice for the simulation of the optical absorption of
electronic materials is time-dependent density functional theory (TDDFT)
grossbook , owing to its high computational efficiency and comparable
accuracy. TDDFT is the most important extension of Kohn-Sham ground-state DFT,
the standard method in electronic structure calculations. The only
approximation made in TDDFT is the dynamical exchange-correlation (XC)
potential, which includes all unknown many-body effects. The simplest
construction is called adiabatic (ad) approximation zs , which takes the same
form of the static XC potential but replaces the ground-state density
$n_{0}({\bf r})$ with the instantaneous time-dependent density $n({\bf r},t)$:
$v_{\rm xc}^{ad}([n];{\bf r},t)=\delta E_{\rm xc}[n_{0}]/\delta n_{0}({\bf
r})|_{n_{0}({\bf r})=n({\bf r},t)}~{}.$ The advantage of this approach is its
simplicity in both theoretical construction and numerical implementation.
Although the adiabatic TDDFT cannot properly describe multiple excitations, it
has become the most popular approach in the study of low-lying single-particle
excitations (i.e., only one electron in the excitated states).

Figure 1: Chemical structures of the computationally studied light-emitting
congugated polymers. Table 1: Excitation energies of singlet-singlet
($S_{0}-S_{1}$) and singlet-triplet ($S_{0}-T_{1}$) gaps (in units of eV) of
polymers of length of $\sim 10~{}{\rm nm}$ in gas phase calculated using the
adiabatic TDDFT methods with the ground-state geometries optimized on the
respective density functionals. Basis set 6-31G is used in all calculations.
The number in parentheses is the number of rings included in our calculations.
1 hartree = 27.21 eV.

| $S_{0}-S_{1}$ |  | $S_{0}-T_{1}^{a}$
---|---|---|---
Polymer | Expta | LSDA | TPSS | TPSSh | B3LYP | PBE0 |  | Expta | LSDA | TPSS | TPSSh | B3LYP | PBE0
P3OT(28) | 2.8-3.8 | $0.99$ | $0.99$ | $1.35$ | $1.59$ | $1.76$ |  | 1.7-2.2 | $0.90$ | $0.80$ | $0.88$ | $0.96$ | $0.95$
PBOPT(32) | 2.52 | $1.49$ | $1.55$ | $1.96$ | $2.26$ | $2.39$ |  | 1.60 | $1.37$ | $1.31$ | $1.42$ | $1.57$ | $1.54$
MEHPPV(16) | 2.48 | $1.14$ | $1.27$ | $1.66$ | $1.94$ | $2.07$ |  | 1.30 | $1.04$ | $1.08$ | $1.18$ | $1.31$ | $1.24$
PFO(36) | 3.22 | $2.30$ | $2.45$ | $2.89$ | $3.13$ | $3.30$ |  | 2.30 | $2.22$ | $2.23$ | $2.34$ | $2.45$ | $2.43$
DHOPPV(16) | 2.58 | $1.14$ | $1.27$ | $1.67$ | $1.95$ | $2.07$ |  | 1.50 | $1.04$ | $1.08$ | $1.18$ | $1.32$ | $1.24$
PPY(24) | 3.4-3.9 | $1.82$ | $2.10$ | $2.61$ | $2.87$ | $3.03$ |  | 2.4-2.5 | $1.82$ | $1.99$ | $2.11$ | $2.23$ | $2.20$
CN-MEHPPV(16) | 2.72 | $1.10$ | $1.34$ | $1.84$ | $2.16$ | $2.27$ |  | N/A | $1.06$ | $1.22$ | $1.34$ | $1.48$ | $1.43$
PANi(20) | 2.00 | $2.34$ | $2.53$ | $3.05$ | $3.30$ | $3.44$ |  | $<0.9$ | $2.31$ | $2.43$ | $2.63$ | $2.75$ | $2.73$

aFrom Ref. monkman01 , in which there is a small red shift in gas phase,
compared to those in solvent (see discussion in the context). bNotation of
Ref. birks  is used. Note that all the groups of -(CH2)nCH3 in polymers have
been replaced with the hydrogen (-H).

Our previous studies of small molecules ttz082  and molecular materials tt09
show that the excitation energies obtained with the adiabatic TDDFT agree
fairly well with experiments. In the present work, we calculate the lowest
singlet-singlet ($S_{0}-S_{1}$) and singlet-triplet ($S_{0}-T_{1}$) excitation
energies of a series of light-emitting organic conjugated polymers (see Fig. 1
for their chemical structures). The singlet-singlet excitation is responsible
for the strong ultraviolet (UV) or near-UV optical absorption, while the
singlet-triplet excitation is responsible for weak fluorescence. Our
calculations show that, when the dihedral angles note1  between two adjacent
phenyl rings obtained by the geometry optimization on ground-state DFT methods
are close to experimental dihedral angles, the calculated TDDFT excitation
energies agree well with experiments, regardless of whether the excitations
arise from singlet-singlet excitations or singlet-triplet excitations. This
suggests that in TDDFT calculations, there are two sources of error. One is
from the adiabatic approximation itself note2 , and the other, much larger
than the first one, arises from inaccuracy of the ground-state DFT geometries.
In order to identify these errors, here we employ five commonly-used density
functionals. Two of them, the local spin density approximation (LSDA) and the
meta-generalized gradient approximation (meta-GGA) of Tao, Perdew, Staroverov,
and Scuseria (TPSS) tpss , are pure density functionals, while the other
three, TPSSh sstp1  (a hybrid of the TPSS meta-GGA with $10\%$ exact
exchange), B3LYP b3lyp  (a hybrid with $20\%$ exact exchange), and PBE0 gus99
(a hybrid of the Perdew-Burke-Ernzerhof (PBE) pbe96  GGA with $25\%$ exact
exchange) are hybrid functionals with increasing amount of exact exchange from
TPSSh, B3LYP to PBE0.

Moreover, in the simulation of electronic excitations of small molecules and
molecular materials, the most effort has been devoted to the study of the
absorption arising from singlet-singlet excitation, leaving the singlet-
triplet excitation less investigated perun . An important reason for this
omission is that triplet-state energies are not easy to measure through direct
optical absorption due to very low singlet-triplet ($S_{0}-T_{1}$) absorption
coefficient walters  and low phosphorescence quantum yield roman
($<10^{-6}$). The major approaches to probe triplet states in conjugated
polymers are the charge recombination or energy transfer, and singlet-triplet
($T_{1}-S_{0}$ or $S_{1}-T_{1}$) intersystem crossing reindl ; parker ; birks
. The observation of $T_{1}-S_{0}$ phosphorescence from molecules initially
excited into $S_{1}$ is clear evidence for a radiationless transition from
$S_{1}$ to an isoenergetic level of the triplet manifold, corresponding to
singlet-triplet intersystem crossing. Singlet-triplet intersystem crossing can
occur either from the zero-point vibrational level of $S_{1}$ or from
thermally-populated vibrational level of $S_{1}$ into an excited vibrational
level of $T_{1}$, or more probably into a higher excited triplet state
$T_{2}$, which is closer in energy to $S_{1}$. It has been found burrows01 ;
burrows02  that the properties of the triplet states directly impact device
performance. For example, the formation of triplet states may cause the loss
of the device efficiency in these materials and thus can limit device
performance and operational life span. Therefore, investigation of triplet
excitations is crucial for a full understanding of electroluminescence
behavior of conjugated organic polymers and for the improvement of new
materials.

Monkman and collaborators burrows02 ; monkman01  investigated the photophysics
of triplet states in a series of conjugated polymers and measured the
excitation energies of the lowest singlet- and triplet-excitated states. Their
measurements show that the excitation energies in general respect the well-
known rule of thumb found for small molecules:

$\displaystyle E_{T}\approx 2E_{S}/3,$ (1)

where $E_{T}$ is the triplet excitation energy and $E_{S}$ is the singlet-
singlet excitation energy. As a second part of our work, we calculate the
singlet-triplet excitation energies of the polymers with the adiabatic TDDFT.
We find that, without exact exchange mixing, a pure semilocal density
functional cannot satisfy the thumb rule of Eq. (1), suggesting inadequacy of
the adiabatic semilocal functionals in predicting the triplet excitation
energies for polymers.

Table 2: Torsions of the conjugated polymers Polymer | Expt | PBE0 | Energy
---|---|---|---
P3OT | $\sim 24\,^{\circ}$ | $\sim 0\,^{\circ}$ | red shift
PBOPT | $\sim 35\,^{\circ}$ | $\sim 40\,^{\circ}$ | On experiment
MEHPPV | $\sim 20\,^{\circ}$ | $\sim 1\,^{\circ}$ | red shift
PFO | $\sim 40\,^{\circ}$ | $\sim 38\,^{\circ}$ | On experiment
DHOPPV | $\sim 20\,^{\circ}$ | $\sim 0\,^{\circ}$ | red shift
PPY | $\gtrsim 0\,^{\circ}$ | $\sim 0-1\,^{\circ}$ | slightly red shift
CN-MEHPPV | $\sim 20\,^{\circ}$ | $\sim 0\,^{\circ}$ | red shift
PANi | $\sim 0\,^{\circ}$ | $\sim 18-26\,^{\circ}$ | too blue shift
Table 3: Excitation energies of singlet-singlet ($S_{0}-S_{1}$) and singlet-
triplet ($S_{0}-T_{1}$) gaps (in units of eV) of polymers of length of $\sim
10~{}{\rm nm}$ in benzene solution calculated using the adiabatic TDDFT
methods with the ground-state geometries optimized on the respective density
functionals. The solvent effects are taken into account through PCM
(polarizable continuum model) method. Basis set 6-31G is used in all
calculations. The number in parentheses is the number of rings included in our
calculations. 1 hartree = 27.21 eV.

| $S_{0}-S_{1}$ |  | $S_{0}-T_{1}^{b}$
---|---|---|---
Polymer | Expta | LSD | TPSS | TPSSh | B3LYP | PBE0 |  | Expta | LSD | TPSS | TPSSh | B3LYP | PBE0
P3OT(28) | 2.8-3.8 | $0.97$ | $0.97$ | $1.32$ | $1.56$ | $1.73$ |  | 1.7-2.2 | $0.89$ | $0.80$ | $0.87$ | $0.95$ | $0.94$
PBOPT(32) | 2.52 |  |  |  |  |  |  | 1.60 |  |  |  |  |
MEHPPV(16) | 2.48 | $1.12$ | $1.25$ | $1.64$ | $1.91$ | $2.04$ |  | 1.30 | $1.03$ | $1.07$ | $1.18$ | $1.32$ | $1.25$
PFO(36) | 3.22 | $2.30$ | $2.45$ | $2.88$ | $3.12$ | $3.29$ |  | 2.30 | $2.22$ | $2.24$ | $2.35$ | $2.46$ | $2.43$
DHOPPV(16) | 2.58 | $1.12$ | $1.25$ | $1.64$ | $1.92$ | $2.04$ |  | 1.50 | $1.03$ | $1.07$ | $1.18$ | $1.32$ | $1.25$
PPY(24) | 3.4-3.9 | $2.08$ | $2.16$ | $2.61$ | $2.85$ | $3.01$ |  | 2.4-2.5 | $2.02$ | $1.99$ | $2.11$ | $2.23$ | $2.20$
CN-MEHPPV(16) | 2.72 | $1.10$ | $1.32$ | $1.80$ | $2.10$ | $2.21$ |  | N/A | $1.05$ | $1.21$ | $1.34$ | $1.48$ | $1.43$
PANi(20) | 2.00 | $2.33$ | $2.53$ | $3.03$ | $3.27$ | $3.41$ |  | $<0.9$ | $2.30$ | $2.42$ | $2.62$ | $2.75$ | $2.73$

aFrom Ref. monkman01 . bNotation of Ref. birks  is used. Note that all the
groups of -(CH2)nCH3 in polymers have been replaced with the hydrogen (-H).

Computational method: All our calculations were performed on the molecular-
structure code Gaussian 03 g03 . The initial geometries are prepared with
GaussView 4, while the dihedral angles are manually adjusted to be $\sim
30\,^{\circ}$. Then we optimize the geometries on respective ground-state DFT
methods. Finally we calculate the excitation energies from the optimized
ground-state geometries with the adiabatic TDDFT density functionals. For
consistency, basis set 6-31G was used in both ground-state and time-dependent
DFT calculations. In order to check whether our conclusion is affected by the
choice of basis set, we repeat our calculations for polymer P3OT using a
larger basis set 6-31G(d) that has diffusion functions. Our calculations show
that the excitation energy obtained with 6-31G(d) is larger only by $<0.2$ eV
than that obtained with 6-31G basis set. The excitation energies of the
polymers in benzene solvent are calculated with PCM (polarizable continuum
model) cmt97 . The polymers we study here have chain length of $\sim 10$ nm.
Since the groups of -(CH2)nCH3 only has little effect upon the properties of
the polymers ttz082 , these groups have been removed from the backbone of a
polymer and are, therefore, excluded in all calculations.

Table 1 shows the first singlet and triplet excitation energies of the
polymers in gas phase calculated with the adiabatic TDDFT. The experimental
results are also listed for comparison. Usually a polymer is of infinite chain
length. In practical calculations, we only choose several repeating monomeric
units. The number of “molecular” rings included in our calculations for each
polymer is given in the parentheses in Tables 1 and 3. These numbers are
chosen so that the lengths of the polymers are about 10 nm. This size effect
will be reduced by increasing the repeating units. However, adding the
repeating units will simultaneously increase the computational time. On the
other hand, high accuracy usually can be achieved by using large basis set,
which will result in significant increase in computational time. In the
present calculations, we use a basis set which is relatively smaller than
those used in small molecular calculations, and prepare the polymers with
moderate length of chain. This is a balanced choice between the size effect
and the accuracy we can tolerate.

From Table 1 we observe that, among the five adiabatic TDDFT methods, the
adiabatic PBE0 functional yields the most accurate excitation energies. This
is consistent with our previous studies ttz082 ; tt09 . We can see from Table
1 that the difference between the singlet and the triplet excitation energies,
$E_{S}-E_{T}$, is $\sim 0-0.1$ eV for LSDA, $\sim 0.1-0.2$ eV for meta-GGA,
$\sim 0.5$ eV for TPSSh, $\sim 0.6$ eV for B3LYP, and $\sim 0.8$ eV for PBE0.
The difference increases as the amount of exact exchange increases. However,
some studies suggest itc05 ; itc07  that for semilocal density functionals
(LSDA, GGA, and meta-GGA), this difference may vanish in the limit of infinite
chain length, a result similar to the performance of semilocal functionals for
solids. Mixing exact exchange into a semilocal functional will partly correct
the errors from self interaction, improve the asymptotic behavior of the XC
potential, and build in other many-body properties such as excitonic effects
itc05 ; itc07  which have not been taken into account properly in pure density
functional approximations and thus will lead to a finite difference in this
limit.

Interestingly, we find that when the theoretical dihedral angle is smaller
than the experimental dihedral angle, the TDDFT methods tend to underestimate
the excitation energies regardless of whether the excitation is singlet or
triplet. When the theoretical dihedral angle is close to the experimental one,
the TDDFT excitation energies are in good agreement with experiments. Our
calculations show that in rare cases, theoretical dihedral angles can be
greater that experimental estimates. In this case, the excitation energies are
overestimated by the TDDFT methods. A comparison of the dihedral angles
between theoretical and experimental estimates is displayed in Table 2. The
origin of torsional angles (or generally tortional disorder) of polymers is
complicated. It may arise from interchain interaction in amorphous polymeric
materials sergei1 ; sergei2  or from van der Waals interaction dion ;
scheffler  between phenyl rings. These effects have not been properly taken
into account in current DFT methods.

The excitation energies of the polymers in benzene solvent are summarized in
Table 3. From Table 3, we can see that the lowest singlet-singlet excitation
energies in solution have a red shift of $\sim 0.01-0.05$ eV, compared to
those in gas phase (Table 1). This is consistent with what we have observed
for oligomers ttz082 ; tt09 . However, this trend does not apply to the
triplet excitation. Triplet excitation energies are nearly the same whether
the polymer is in gas phase or in solution.

In conclusion, we have investigated the lowest excitation energies of several
light-emitting conjugated polymers from the adiabatic TDDFT methods. Our
calculations show that the calculated excitation energies are in good
aggrement with experiments only when the theoretical torsions agree with
experimental estimates. If the theoretical dihedral angles are smaller than
the experiments, the TDDFT excitation energies tend to be underestimated. If
the theoretical dihedral angles are greater than the experiments, as in rare
case, the TDDFT excitation energies tend to be overestimated. Furthermore, we
find that, a semilocal functional without exact exchange mixing does not
satisfy the well-known “two-third” thumb rule relation between the singlet-
singlet and singlet-triplet excitation energies. For semilocal functionals,
the difference in energy between singlet state and triplet state is less than
0.1 eV for polymers with chain length of 10 nm and may vanish in the limit of
infinite chain length. Compared to semilocal functionals, hybrid functionals
yield much larger difference between singlet-singlet and singlet-triplet
excitation energies for polymers with finite chain length as well as with
infinite chain length. This difference increases with more exact exchange
mixed in semilocal functionals, and is nonzero even in the limit of infinite
chain length.

###### Acknowledgements.

The authors thank Richard Martin and John Perdew for valuable discussion and
suggestions. This work was carried out under the auspices of the National
Nuclear Security Administration of the U.S. Department of Energy at Los Alamos
National Laboratory under Contract No. DE-AC52-06NA25396, and was supported by
the LANL LDRD program.

## References

  * (1) A. Kraft, A.C. Grimsdale, and A.B. Holmes, Angew. Chem. Int. Ed. 37, 402 (1998); T. Jüstel, H. Nikol, and C. Ronda, Angew. Chem. Int. Ed. 37, 3084 (1998); U. Mitschke and P. Bäuerle, J. Mater. Chem. 10, 1471 (2000); D.Y. Kim, H.N. Cho, and C.Y. Kim, in Progress in Polymer Science, v. 25 (Elsevier Science Ltd. 2000).
  * (2) P. May, Phys. World 8, 52 (1995).
  * (3) Time-Dependent Density Functional Theory, Lecture Notes in Physics, Vol. 706, edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross (Springer, Berlin, 2006).
  * (4) A. Zangwill and P. Soven, Phys. Rev. Lett. 45, 204 (1980).
  * (5) J. Tao, S. Tretiak, and J.-X. Zhu, J. Phys. Chem. 112, 13701 (2008).
  * (6) J. Tao and S. Tretiak, J. Chem. Theory Comput. 5, 866 (2009).
  * (7) Dihedral angle is defined as the angle formed by two planes containing two adjacent phenyl or other “molecular” rings.
  * (8) It has been shown that the largest source of error in the prediction of low-lying excitation energies arises from the approximation to the static XC potential [S.J.A. van Gisbergen, F. Kootstra, P.R.T. Schipper, O.V. Gritsenko, J.G. Snijders, and E.J. Baerends, J. Chem. Phys. 109, Phys. Rev. A 57, 2556 (1998)].
  * (9) J. Tao, J.P. Perdew, V.N. Staroverov, and G.E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003).
  * (10) V.N. Staroverov, G.E. Scuseria, J.Tao, and J.P. Perdew, J. Chem. Phys. 119, 12129 (2003).
  * (11) P.J. Stephens, F.J. Devlin, C.F. Chabalowski, and M.J. Frisch, J. Phys. Chem. 1994, 98, 11623.
  * (12) C. Adamo, G.E. Scuseria, and V. Barone, J. Chem. Phys. 111, 2889 (1999); M. Ernzerhof and G.E. Scuseria, J. Chem. Phys. 1999, 110, 5029.
  * (13) J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * (14) S. Perun, J. Tatchen, and C.M. Marian, ChemPhysChem 9, 282 (2008).
  * (15) K.A. Walters, K.D. Ley, and K.S. Shanze, Chem. Commun. 10, 1115 (1998).
  * (16) Yu. V Romanovskii, A. Gerhard, B. Schweitzer, U. Scherf, R.I. Personov, and H. Bässler, Phys. Rev. Lett. 84, 1027 (2000).
  * (17) S. Reindl and A. Penzkofer, Chem. Letts. 211, 431 (1996).
  * (18) C.A. Parker, Photoluminescence of solutions (Elsevier, Amsterdam, 1968).
  * (19) J.B. Birks, Photophysics of aromatic molecules (Wiley-Interscience, London, 1970).
  * (20) H.D. Burrows, J. Seixas de Melo, C. Serpa, L.G. Arnaut, A.P. Monkman, I. Hamblett, and S. Navaratnam, J. Chem. Phys. 115, 9601 (2001).
  * (21) H.D. Burrows, J. Seixas de Melo, C. Serpa, L.G. Arnaut, M. da G. Miguel, A.P. Monkman, I. Hamblett, and S. Navaratnam, Chem. Letts. 285, 3 (2002).
  * (22) A.P. Monkman, H.D. Burrows, L.J. Hartwell, L.E. Horsburgh, I. Hamblett, and S. Navaratnam, Phys. Rev. Lett. 86, 1358 (2001).
  * (23) M.J. Frisch et al., Gaussian, Inc., Pittsburgh PA (2003).
  * (24) E. Cancès, B. Mennucci, and J. Tomasi, J. Chem. Phys. 107, 3032 (1997).
  * (25) S. Tretiak, K. Igumenshchev and V. Chernyak, Phys. Rev. B 71, 033201 (2005).
  * (26) K. Igumenshchev, S. Tretiak, and V. Chernyak, J. Chem. Phys. 127, 114902 (2007).
  * (27) P. Yang, E.R. Batista, S. Tretiak, A. Saxena, R.L. Martin, and D.L. Smith, Phys. Rev. B 76, 241201(R) (2007).
  * (28) S. Kilina, E.R. Batista, P. Yang, S. Tretiak, A. Saxena, R.L. Martin, and D.L. Smith, ACS Nano 2, 1381 (2008).
  * (29) M. Dion, H. Rydberg, E. Schroder, D.C. Langreth, and B.I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).
  * (30) A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009).


Sanaz Pooya and Hang Wang

††footnotetext:  MSC classification: 46L80, 19D55, 20F65

Abstract. We provide an explicit description of the K-classes of higher
Kazhdan projections in degrees greater than 0 for specific free product groups
and Cartesian product groups. Employing this description, we obtain new
calculations of Lott’s delocalised $\ell^{2}$-Betti numbers. Notably, we
establish the first non-vanishing results for infinite groups.

##  1 Introduction

Within operator algebras K-theory holds importance as it encapsulates
extensive information regarding a $\mathrm{C}^{*}$-algebra. In view of its
richness in information, computational challenges persist. In light of these
challenges, understanding specific K-classes can provide us with partial
information that may lead to improved understanding of an object it is
associated with. In the framework of group $\mathrm{C}^{*}$-algebras there are
numerous ways to produce K-classes that can be analysed. The averaging
projection in the $\mathrm{C}^{*}$-algebra of a finite group is one of the
most elementary such examples. More involved K-classes can be produced as Bott
projections and their various generalisations. Also K-classes arising from
generalised Fredholm operators by applying the Baum-Connes assembly map can to
a certain extend be considered as more concrete than general K-classes of
group $\mathrm{C}^{*}$-algebras. A rather different kind of K-classes was
recently introduced under the name of higher Kazhdan projections in [LNP20].
They can provide a new way to obtain concrete K-classes. Their construction
was motivated by that of Kazhdan projections associated to property (T)
groups. However, in the generality in which they were defined, they can also
be defined for non-property (T) groups. Let us briefly recall the construction
and discuss to what extent these classes could be understood so far. Let $G$
be a discrete group of type $F_{n+1}$. For a suitable unitary representation
$(\pi,\mathcal{H})$ of $G$, the first author, Li and Nowak defined a sequence
of higher Kazhdan projections $(p_{n})_{n\geq 0}$. They lie in matrices over
$\mathcal{B}(\mathcal{H})$. By construction, $p_{n}$ is the projection onto
harmonic $n$-cochains in a suitable chain complex computing cohomology of $G$
with coefficients in $(\pi,\mathcal{H})$, which relates it directly to reduced
group cohomology. In order to view $p_{n}$ inside matrices over
$\mathrm{C}^{*}_{\pi}(G)$ we need a spectral gap for the $n$-th Laplacian
$\Delta_{n}$. This is equivalent to that the $n$-th and $n+1$-st cohomology
with coefficient in $\mathcal{H}$ are reduced [BN15]. If $\pi$ is the
universal representation of a property (T) group $G$, then
$p_{0}\in\mathrm{C}^{*}_{\mathrm{max}}(G)$ is the classical Kazhdan
projection. The case where $\pi$ is chosen to be the left regular
representation $\lambda$ is particularly interesting, since in this case the
trace of the K-class of $p_{n}$ is the $n$-th $\ell^{2}$-Betti number of the
group. The examples of non-abelian free groups $\mathbb{F}_{k}$ and
$\mathrm{PSL}(2,\mathbb{Z})$ showcase different behavior of $p_{1}$. In the
example of $\mathbb{F}_{k}$, we have that $[p_{1}]=(k-1)[1]$ in K-theory,
while for $\mathrm{PSL}(2,\mathbb{Z})$ we have that
$[p_{1}]\notin\mathbb{Z}[1]$ due to the non-integral first $\ell^{2}$-Betti
number of the group. In contrast to the classical Kazhdan projection which
only exists when $G$ has property (T) we now saw that higher Kazhdan
projections exist and exhibit different behavior in this more general setting.
A natural question to ask is whether it is possible to explicitly describe
their K-classes. In this article we answer this question positively by
explicitly describing the K-class of higher Kazhdan projections in two classes
of groups: free products of finite cyclic groups including
$\mathrm{PSL(2,\mathbb{Z})}$ and certain Cartesian products. By directly
computing the space of harmonic $n$-cochains we are able to obtain very
concrete representatives for the K-class of higher Kazhdan projections
associated with these groups.

######  Theorem A.

(Theorem 4.3) Let $G=\mathbb{Z}_{m}\ast\mathbb{Z}_{n}$ for $m\geq 2$ and
$n\geq 3$. The K-class of first higher Kazhdan projection $p_{1}$ in
$\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$ is described by

$[p_{1}]=[1]-\left[\frac{1}{m}\sum_{0\leq
i<m}{s^{i}}\right]-\left[\frac{1}{n}{\sum_{0\leq j<n}{t^{j}}}\right],$

where $s$ and $t$ are generators (of order $m$ and $n$) of the cyclic groups,
respectively.

Employing a product construction, we can produce higher Kazhdan projections
with non-integral K-class in higher degrees. This is the content of the next
theorem.

######  Theorem B.

(Theorem 4.13) Let $F$ be a finite group let $n\in\mathbb{N}$ and consider the
product $G=\mathbb{F}_{2}\times\dotsc\times\mathbb{F}_{2}\times F$ of $n$
factors of $\mathbb{F}_{2}$ with $F$. Then the K-class of the $n$-th higher
Kazhdan projection $p_{n}$ of $G$ satisfies

$[p_{n}]=\left[\frac{1}{|F|}{\sum_{g\in F}{g}}\right].$

Besides providing a clearer picture of this new class of cohomologically
defined projections, the relevance of explicitly describing their K-classes
stems from their connections to Lott’s delocalised $\ell^{2}$-Betti numbers
for manifolds [Lot99].
For $g\in G$ we denote its conjugacy class $\langle{g}\rangle$. Associated
with the conjugacy class of $g\in G$ is a delocalised trace, which is the
tracial functional, defined as

$\tau_{\langle
g\rangle}\colon\ell^{1}(G)\rightarrow\mathbb{C},\qquad\tau_{\langle
g\rangle}(f)=\sum_{h\in{\langle g\rangle}}{f(h)}.$

Aiming to pair such delocalised traces with K-classes an obstruction can be
observed for infinite conjugacy classes when $\tau_{\langle g\rangle}$ does
not necessarily extend to the $\mathrm{C}^{*}$-algebra closure in question.
This problem has different remedies. Overcoming this problem Lott introduced
his delocalised $\ell^{2}$-Betti numbers as the limit of delocalised traces of
heat kernel operators for closed manifolds, whose analogue for groups lies in
matrices over $\ell^{1}(G)$. More concretely, for a group $G$ denoting by
$\Delta_{n}$ the $n$-th combinatorial Laplacian, the $n$-th delocalised
$\ell^{2}$-Betti number would be described by the following expression:

$\displaystyle\beta^{(2)}_{n,\langle
g\rangle}(G)=\lim_{t\to\infty}\tau_{\langle g\rangle}(e^{-t\Delta_{n}}).$

However, in view of the context of higher Kazhdan projections it makes sense
to adopt a more K-theoretic definition. Since
$\lim_{t\to\infty}e^{-t\Delta_{n}}$ considered as a strong limit does not
always belong to $\ell^{1}(G)$, one would need to find a different subalgebra
of $\mathrm{C}^{*}_{\text{red}}(G)$ to which these traces can extend. When $G$
is a hyperbolic group, Puschnigg has constructed suitable smooth dense
subalgebras of $\mathrm{C}^{*}_{\text{red}}(G)$, which have the same K-theory
as $\mathrm{C}^{*}_{\text{red}}(G)$ and to which delocalised traces extend
(see [Pus10]). For the purpose of this article we then adopt the following
definition.

######  Definition C.

(Definition 2.11) Let $G$ be a group of type $F_{n+1}$, and $g\in G$. Assume
that there is a smooth subalgebra
$\mathcal{S}\subset\mathrm{C}^{*}_{\text{red}}(G)$ to which the delocalised
trace $\tau_{\langle g\rangle}$ extends. Assume further that the K-class of
$p_{n}$ lies in $\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$. We define
the $n$-th delocalised $\ell^{2}$-Betti number of $G$ as

$\displaystyle\beta^{(2)}_{n,\langle g\rangle}(G)=\tau_{\langle
g\rangle}([p_{n}]).$

As a byproduct of the explicit description of higher Kazhdan projections, we
establish the first calculations for non-zero delocalised $\ell^{2}$-Betti
numbers of infinite groups.

######  Corollary D.

(Corollary 4.10) The delocalised $\ell^{2}$-Betti numbers for the group
$G=\mathbb{Z}_{m}\ast\mathbb{Z}_{n}$ are

$\beta^{(2)}_{1,\langle{g}\rangle}(G)=\begin{cases}1-\frac{1}{m}-\frac{1}{n}&\qquad
g=e\\\ -\frac{|\langle{g}\rangle|}{m}&\qquad
g\in\mathbb{Z}_{m}\setminus\\{e\\}\\\ -\frac{|\langle{g}\rangle|}{n}&\qquad
g\in\mathbb{Z}_{n}\setminus\\{e\\}\\\
0\qquad&\qquad\text{otherwise}\end{cases}$

and $\beta^{(2)}_{k,\langle{g}\rangle}(G)=0$ for $k\neq 1$ and all $g\in G$.

In addition to establishing non-vanishing results, we also demonstrate several
vanishing instances. Notably, we prove that if the $n$-th $\ell^{2}$-Betti
number of a hyperbolic group $G$ vanishes, then the $n$-th delocalised
$\ell^{2}$-Betti numbers $\beta^{2}_{n,\langle g\rangle}(G)$ vanishes for all
$g\in G$ (see Proposition 3.2). Moreover, we provide more evidences concerning
an observation made by Lott on vanishing of delocalised $\ell^{2}$-Betti
numbers in the case of torsion free hyperbolic groups. Specifically, we
establish that for such a group $G$ and under the presence of a spectral gap
for $\Delta_{n}$, all $n$-th delocalised $\ell^{2}$-Betti numbers
$\beta^{2}_{n,\langle g\rangle}(G)$ vanish for $g\neq e$ (see Proposition
3.5). This article contains four sections. After the introduction, in Section
2 we collect preliminaries on higher Kazhdan projections, delocalised
$\ell^{2}$-Betti numbers and smooth subalgebras. In Section 3, we prove our
vanishing results. In Section 4 we prove Theorem A and compute its first
delocalised $\ell^{2}$-Betti numbers as it is stated in Corollary D. Moreover,
we prove Theorem B.

### Acknowledgments

H.W. is supported by the grants NSFC 12271165,23JC1401900 and in part by
Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014).
S.P. was supported by Knut and Alice Wallenberg foundation through grant
number 31001288. We would like to thank John Lott and Aaron Tikuisis for very
useful comments and Rufus Willett for various conversations around the topic.

##  2 Preliminaries

###  2.1 Higher Kazhdan projections

In this section we briefly recall the construction of higher Kazhdan
projections from [LNP20]. Let $G$ be a group of type $F_{n+1}$ with a chosen
model $X$ of $\mathrm{B}G$ with finite $n+1$-skeleton. Let $(\pi,\mathcal{H})$
be a unitary representation of $G$. We consider the group cohomology of $G$
with coefficients in $\mathcal{H}$ computed via

$\mathrm{C}^{0}(G,\mathcal{H})\stackrel{{\scriptstyle
d_{0}}}{{\longrightarrow}}\mathrm{C}^{1}(G,\mathcal{H})\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d_{i-2}}}{{\longrightarrow}}\mathrm{C}^{i-1}(G,\mathcal{H})\stackrel{{\scriptstyle
d_{i-1}}}{{\longrightarrow}}\mathrm{C}^{i}(G,\mathcal{H})\stackrel{{\scriptstyle
d_{i}}}{{\longrightarrow}}\mathrm{C}^{i+1}(G,\mathcal{H})\stackrel{{\scriptstyle
d_{i+1}}}{{\longrightarrow}}\cdots.$

By the assumption that $G$ is of type $F_{n+1}$ we may consider the following
identification when $i\leq n$

$\mathrm{C}^{i}(G,\mathcal{H})=\left\\{f\colon
X^{(i)}\rightarrow\mathcal{H}\right\\}\cong\mathcal{H}^{\oplus k_{i}},$

with $k_{i}$ being the number of cells in $X^{(i)}$. The coboundaries
$d_{i}$’s are identified with matrices in $\mathrm{M}_{k_{i+1}\times
k_{i}}(\mathbb{Z}G)$ over the integral group ring $\mathbb{Z}G$. The
(combinatorial) Laplacian in degree $n$ is the operator

$\Delta_{i}=d_{i}^{*}d_{i}+d_{i-1}d^{*}_{i-1}\in\mathrm{M}_{i_{k}}(\mathbb{Z}G).$

The kernel of $\pi(\Delta_{i})$ is the space of harmonic $i$-cochains for
$\pi$, which is by the Hodge-de Rham isomorophism identified with the reduced
cohomology ${\overline{\mathrm{H}}}^{n}(G,\mathcal{H})$, that is

$\ker\pi(\Delta_{i})\simeq{\overline{\mathrm{H}}}^{i}(G,\mathcal{H}).$

Recall that in the setting where the cochain spaces in a cochain complex are
equipped with a Banach space structure the corresponding reduced cohomology
group are defined by the quotients
${\overline{\mathrm{H}}}^{i}(G,\mathcal{H}):=\ker d_{i}\leavevmode\nobreak\
/\leavevmode\nobreak\ \overline{\mathrm{im}\,d_{i-1}}$, where
$\overline{\mathrm{im}\,d_{i-1}}$ is the closure of the image of the
codifferential $d_{i-1}$. The cohomology is reduced in degree $i$ if
${\overline{\mathrm{H}}}^{i}(G,\mathcal{H})=\mathrm{H}^{i}(G,\mathcal{H})$.

######  Definition 2.1.

[LNP20] Let $G$ be a group of type $F_{n+1}$ with a chosen model $X$ for
$\mathrm{B}G$. Let $(\pi,\mathcal{H})$ be a unitary representation of $G$. The
projection $p_{n}\colon\mathcal{H}^{\oplus
n_{k}}\rightarrow\ker\pi(\Delta_{n})$ is called a higher Kazhdan projection in
degree $n$, where $n_{k}$ is the number of cells in the $n$-th skeleton of
$X$.

Similar to the situation with property (T), higher Kazhdan projections exist
in the presence of a spectral gap at $0$.

######  Lemma 2.2.

[BN20, Proposition 16.2] Let $(\pi,\mathcal{H})$ be a unitary representation
of $G$. The Laplacian $\pi(\Delta_{n})$ has spectral gap at $0$ if and only if
the cohomology is reduced in degree $n$ and $n+1$.

Assume that for the left regular representation $\lambda$, the operator
$\lambda(\Delta_{n})$ has spectral gap at $0$ so that
$[p_{n}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$. Let $\tau$ denote
the canonical trace on $\mathrm{C}^{*}_{\text{red}}(G)$ and by abuse of
notation its extension to the K-theory level. By [LNP20, Proposition 7], the
$n$-th $\ell^{2}$-Betti numbers of $G$ are obtained by the trace pairing

(2.1) $\displaystyle\beta^{(2)}_{n}(G)=\tau([p_{n}]).$

Convention. We will exclusively work with the left regular representation.
Further, By abuse of notation, we identify $\lambda(\Delta_{n})$ as
$\Delta_{n}$.

###  2.2 Delocalised $\ell^{2}$ -Betti numbers

In [Lot99], John Lott introduced delocalised $\ell^{2}$-Betti numbers as part
of his delocalised $\ell^{2}$-invariants. These are analogue of the
$\ell^{2}$-invariants of closed Riemannian manifolds. Before presenting the
definition, we recall the notion of delocalised traces that will be one of its
ingredients.

######  Definition 2.3.

Let $G$ be a discrete group and $g\in G$. The delocalised trace $\tau_{\langle
g\rangle}$ on $\ell^{1}(G)$ is the bounded linear map with tracial property

$\tau_{\langle g\rangle}:\ell^{1}(G)\rightarrow\mathbb{C},\qquad\tau_{\langle
g\rangle}(f)=\sum_{h\in\langle g\rangle}f(h),$

where $\langle g\rangle$ denotes the conjugacy class of $g$.

Precomposition with matrix traces makes it possible to consider delocalised
traces as functionals on matrix algebras. The following analog is in complete
analogy with Lott’s definition for closed manifolds. The difference is that we
consider groups rather than manifolds and combinatorial Laplacian instead of
Laplace-Baltrami operators.

######  Definition 2.4.

(cf. [Lot99, Definition 3]) Let $G$ be a discrete group. Let $g\in G$, the
$n$-th delocalised $\ell^{2}$-Betti number of $G$ with respect to the
conjugacy class $\langle g\rangle$ is defined by

(2.2) $\displaystyle\beta^{(2)}_{n,\langle
g\rangle}(G)=\lim_{t\to\infty}\tau_{\langle g\rangle}(e^{-t\Delta_{n}}),$

when the limit exists.

We refer to this definition as the analytic definition of delocalised
$\ell^{2}$-Betti numbers. In analogy with Lott’s observation for manifolds in
[Lot99, Proposition 18], self-adjointness of the operator $e^{-t\Delta_{n}}$
together with $\tau_{\langle g\rangle}(a^{*})=\overline{{\tau_{\langle
g^{-1}\rangle}{(a)}}}$ imply that

$\beta^{(2)}_{n,\langle g^{-1}\rangle}(G)=\overline{\beta^{(2)}_{n,\langle
g\rangle}(G)}.$

In some situations our delocalised $\ell^{2}$-Betti numbers arise in Lott’s
setting. Recall that Lott is considering closed manifolds with a normal
$G$-cover to which he associated his invariants. When the classifying space
$\mathrm{B}G$ of a group $G$ has a homotopy type of a closed manifold we can
consider its universal cover and it turns out that the associated delocalised
$\ell^{2}$-Betti numbers agree.

######  Remark 2.5.

Lott defined his invariants with respect to conjugacy class of non-trivial
elements, which makes sense in view of the choice of name delocalised. For
convenience we include the case of the trivial element because all of our
statements work uniformly.

###  2.3 Smooth subalgebras and K-theory pairing with delocalised traces

For the purpose of this article a smooth subalgebra $\mathcal{S}$ of a
$\mathrm{C}^{*}$-algebra is a dense Banach subalgebra which is closed under
holomorphic functional calculus. An important feature of smooth subalgebras is
that they have the same K-theory as the $\mathrm{C}^{*}$-algebra. For a given
group $G$, we are interested in smooth subalgebras of its reduced group
$\mathrm{C}^{*}$-algebra $\mathrm{C}^{*}_{\text{red}}(G)$. Let $g\in G$. If
delocalised traces extend continuously to a smooth subalgebra of
$\mathrm{C}^{*}_{\text{red}}(G)$, this gives rise to a K-theory pairing by
considering

$\tau_{\langle
g\rangle}:\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))\cong\mathrm{K}_{0}(\mathcal{S})\rightarrow\mathbb{C}.$

The following result is a consequence of work of Puschnigg on traces on
unconditional completions of group rings presented in Section 5 of [Pus10]. In
particular it follows from the initial part of the proof of Theorem 5.2 and
Proposition 5.5 in there.

######  Theorem 2.6.

[Pus10, Section 5] Let $G$ be a hyperbolic group. There exists a smooth
subalgebra of $\mathrm{C}^{*}_{\text{red}}(G)$ to which delocalised traces
extend.

######  Definition 2.7.

Let $G$ be a group. A smooth subalgebra of $\mathrm{C}^{*}_{\text{red}}(G)$
which contain the complex group ring $\mathbb{C}G$ and to which all
delocalised traces extend is called a Puschnigg subalgebra.

Recall that for a group $G$ of type $F_{n+1}$, its Laplacian $\Delta_{n}$
belongs to some matrices over $\mathbb{C}G\subseteq\mathcal{S}$. The next
proposition demonstrates that the family of heat operators converges already
in the Puschnigg subalgebras. The convergence is crucial for our later
applications. Our proof is analogous to an argument given in the proof of
[Lot99, Lemma 4].

######  Remark 2.8.

Hyperbolic groups are of type $F_{\infty}$. Therefore we will drop the
assumption of type $F_{n+1}$ when stating our results concerning $p_{n}$, for
$n\in\mathbb{N}$.

######  Proposition 2.9.

Let $G$ be a hyperbolic group. If $\Delta_{n}$ has a spectral gap, then
$p_{n}\in\mathcal{S}$ and

$\lim_{t\to\infty}e^{-t\Delta_{n}}=p_{n}$

in $\mathcal{S}$ for any Puschnigg subalgebra
$\mathcal{S}\subseteq\mathrm{C}^{*}_{\text{red}}(G)$.

###### Proof.

The membership of the heat operator $e^{-t\Delta_{n}}$ in any Banach algebraic
closure of the group ring $\mathbb{C}G$, and in particular to $\mathcal{S}$ is
straightforward to verify. Furthermore, an easy application of holomorphic
functional calculus implies that $p_{n}$ belongs to $\mathcal{S}$. Due to the
fact that $\mathcal{S}$ is holomorphically closed, the following computation
can be done in $\mathcal{S}$. Consider the decomposition

$\displaystyle p_{n}-e^{-t\Delta_{n}}$
$\displaystyle=p_{n}(p_{n}-e^{-t\Delta_{n}})+(id-
p_{n})(p_{n}-e^{-t\Delta_{n}})$ $\displaystyle=(id-p_{n})e^{-t\Delta_{n}}(id-
p_{n})$ $\displaystyle=e^{(id-p_{n})-t\Delta_{n}(id-p_{n})}$
$\displaystyle:=e^{-t\tilde{\Delta_{n}}}.$

We show that $\|e^{-t\tilde{\Delta_{n}}}\|$ goes to zero. Consider the family
of one parameter semi groups $\\{e^{-t\tilde{\Delta_{n}}}\\}_{t>0}$ in the
Banach algebra $\mathcal{S}$. By [Dav80, Theorem 1.22], the limit below exists

$r=\lim_{t\to\infty}t^{-1}\ln\|e^{-t\tilde{\Delta_{n}}}\|,$

and we have that for all $t>0$ the spectral radius of
$e^{-t\tilde{\Delta_{n}}}$ is $e^{rt}$. Let $\lambda_{0}>0$ be the inf of the
spectrum of $\tilde{\Delta_{n}}$. Then the spectral radius of
$e^{-t\tilde{\Delta_{n}}}$ is $e^{-t\lambda_{0}}$. Since $r<0$, then $r/2>r$,
implying that for $t>t_{0}$

$t^{-1}\ln\|e^{-t\tilde{\Delta_{n}}}\|\leq r/2.$

When $t>1$ we obtain that for $C>0$

$\|e^{-t\tilde{\Delta_{n}}}\|\leq Ce^{-t\lambda_{0}/2}.$

Letting $t$ go to infinity, we obtain the desired convergence in
$\mathcal{S}$.

######  Corollary 2.10.

Let $G$ be a hyperbolic group. If $\Delta_{n}$ has spectral gap, then the
K-class $[p_{n}]$ belongs to $\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$.

###### Proof.

By hyperbolicity, the reduced group $\mathrm{C}^{*}_{\text{red}}(G)$ contains
a Puschnigg subalgebra $\mathcal{S}$. In the presence of a spectral gap
$p_{n}$ belongs to some matrix over $\mathcal{S}$. Consequently, the K-class
$[p_{n}]$ belongs to
$\mathrm{K}_{0}(\mathcal{S})=\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$.

Assume that for a given group $G$ there is a Puschnigg subalgebra
$\mathcal{S}\subseteq\mathrm{C}^{*}_{\text{red}}(G)$. The advantage of having
$\lim_{t\to\infty}e^{-t\Delta_{n}}=p_{n}$ in $\mathrm{M}_{k_{n}}(\mathcal{S})$
is that we can do K-theory pairings with delocalised traces. In such a
situation we may rewrite (2.2) from Definition 2.4 to obtain an analogous
formula to (2.1) for $\ell^{2}$-Betti numbers.

######  Definition 2.11.

Let $G$ be a group of type $F_{n+1}$ and $g\in G$. Assume that there is a
smooth subalgebra $\mathcal{S}\subseteq\mathrm{C}^{*}_{\text{red}}(G)$ to
which the delocalised trace $\tau_{\langle g\rangle}$ extends. Assume further
that the K-class of $p_{n}$ lies in
$\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$. We define the $n$-th
delocalised $\ell^{2}$-Betti number of $G$ via the following pairing

(2.3) $\displaystyle\beta^{(2)}_{n,\langle g\rangle}(G)=\tau_{\langle
g\rangle}([p_{n}]).$

We refer to the above definition as the K-theoretic definition of delocalised
$\ell^{2}$-Betti numbers. (See (2.2).)

##  3 Vanishing results for delocalised $\ell^{2}$-Betti numbers

In this section, we present our results concerning the vanishing of
delocalised $\ell^{2}$-Betti numbers. These results encompass connections to
property (T) and $\ell^{2}$-Betti numbers. Furthermore, we provide additional
evidence regarding a remark that Lott made concerning the vanishing of the
invariant for torsion free groups.

######  Remark 3.1.

The vanishing results in Proposition 3.2, Corollary 3.3, and Proposition 3.4
are obtained by appealing to our K-theoretic definition of delocalised
$\ell^{2}$-Betti numbers (Definition 2.11). While in Proposition 3.2 and
Corollary 3.3 in the presence of a spectral gap for a relevant degree of
Laplacian our K-theoretic definition and the analytic definition of
delocalised $\ell^{2}$-Betti numbers agree, the situation with Proposition 3.4
is different. For amenable groups a spectral gap is not expected, hence such
groups do not fit in the setup of analytically defined delocalised
$\ell^{2}$-Betti numbers. The key ingredient to the above mentioned results is
that $p_{n}$ is zero.

######  Proposition 3.2.

Let $G$ be a hyperbolic group. If $\beta_{n}^{(2)}(G)$ vanishes, then
$\beta_{n,\langle g\rangle}^{(2)}(G)$ vanishes for all $g\in G$.

###### Proof.

If the $n$-th $\ell^{2}$-Betti number of $G$ vanishes, then by (2.1), $p_{n}$
vanishes. This leads to the vanishing of all pairings of $[p_{n}]$ with
delocalised traces. Therefore, all $\beta_{n,\langle g\rangle}^{(2)}(G)$
vanish.

######  Corollary 3.3.

Let $G$ be a hyperbolic property (T) group. Then
$\beta^{(2)}_{1,\langle{g}\rangle}(G)$ vanishes for all $g\in G$.

###### Proof.

Groups with property (T) have vanishing first $\ell^{2}$-cohomology [Del77].
This implies that $\beta_{1}^{(2)}(G)$ vanishes. Proposition 3.2 can be now
applied.

######  Proposition 3.4.

Let $G$ be a virtually nilpotent group of type $F_{n+1}$. The delocalised
$\ell^{2}$-Betti numbers $\beta^{(2)}_{n,\langle{g}\rangle}(G)$ vanish for all
$n\in\mathbb{N}$ and all $g\in G$.

###### Proof.

First note that $L^{2}_{RD}(G)\subseteq\mathrm{C}^{*}_{\text{red}}(G)$ is a
Puschnigg subalgebra. Because infinite amenable groups have vanishing
$\ell^{2}$-Betti numbers [CG86], their higher Kazhdan projections $p_{n}$ and
hence their delocalised $\ell^{2}$-Betti numbers vanish.

The following proposition can be compared with [Lot99, Proposition 9] which
deals with closed oriented hyperbolic manifold.

######  Proposition 3.5.

Let $G$ be a torsion free hyperbolic group. Assume that $\Delta_{n}$ has a
spectral gap. Then the $n$-th delocalised $\ell^{2}$-Betti numbers vanish for
all non-trivial $g\in G$.

###### Proof.

The Baum-Connes conjecture holds for hyperbolic groups [Laf02]. Hence, for
$[p_{n}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$, there exists an
abstract elliptic operator whose higher index is $[p_{n}]$. Moreover, given
that $G$ is discrete, one can find via geometric cycles [BHS10] a
representative of the abstract elliptic operator that is $G$-equivariant
elliptic operator $D$ lives on a complete manifold $M$ carrying a proper
cocompact $G$-action.
By the $L^{2}$-Lefschetz fixed point formula, the delocalised trace of the
higher index $\tau_{\langle g\rangle}(\mathrm{ind}_{G}(D))$ has a
cohomological formula over the fixed point submanifold $M^{g}$ (refer to
Definition 5.1 and Theorem 6.1 of [WW16].) The properness of the action
implies that all isotropy subgroups are finite. However, the fact that $G$ is
torsion free implies that the isotropy subgroups are trivial. This indicates
that the action is free, and thus $M^{g}$ is empty since $g\neq e$.
Consequently, the $L^{2}$-Lefschetz number vanishes, resulting in

$\beta_{n,\langle g\rangle}^{(2)}(G)=\tau_{\langle
g\rangle}([p_{n}])=\tau_{\langle g\rangle}(\mathrm{ind}_{G}(D))=0.$

The previous proposition confirms Lott’s speculation concerning vanishing of
delocalised $\ell^{2}$-Betti numbers for discrete torsion free groups. See
[Lot99, page 6].

######  Remark 3.6.

In the proof of Proposition 3.5, we used vanishing of the $L^{2}$-Lefschetz
number $\chi_{g}(G)$. It turns out that the non-vanishing of $\chi_{g}(G)$ for
the Euler operator is an obstruction to the vanishing of the delocalised
$\ell^{2}$-Betti numbers. Indeed, we have

$\chi_{g}(G)=\sum_{n\geq 0}(-1)^{n}\beta^{(2)}_{n,\langle g\rangle}(G).$

If $\chi_{g}(G)$ is not vanishing, then there exists an $n$ such that
$\beta^{(2)}_{n,\langle g\rangle}(G)\neq 0$. This generalises Proposition 11
on page 29 of [Lot99] from the context of finite group to that of a discrete
group with proper cocompact action.

##  4 Computations of higher Kazhdan projections and delocalised
$\ell^{2}$-Betti numbers

In this section we describe the K-theory class of higher Kazhdan projections
$p_{n}$ for $n>0$ in specific classes of groups. The results are obtained
through the identification of the kernel of $\Delta_{n}$. As an application,
we compute the delocalised $\ell^{2}$-Betti numbers. These establish the first
non-vanishing results for infinite groups.

######  Lemma 4.1.

Let $G$ be a virtually free group. Then $p_{1}$ lies in matrices over
$\mathrm{C}^{*}_{\text{red}}(G)$, its K-class belongs to
$\mathrm{K_{0}}(\mathrm{C}^{*}_{\text{red}}(G))$ and it is non-zero. All other
$p_{n}$’s vanish when $n\neq 1$.

###### Proof.

Assume that some free group $\mathbb{F}_{k}$ is a finite index subgroup of
$G$. To establish the presence of a spectral gap, we rely on Lemma 2.2. The
non-amenability of $G$ implies that $\mathrm{H}^{1}(G,\ell^{2}(G))$ is
reduced, and Shapiro lemma implies that $\mathrm{H}^{2}(G,\ell^{2}(G))$ is
reduced (even vanishes) as it is so for $\mathbb{F}_{k}$. Consequently,
$p_{1}$ belongs to matrices over $\mathrm{C}^{*}_{\text{red}}(G)$. Recall
equation 2.1 $\tau([p_{1}])=\beta^{(2)}_{1}(G)$. Since $\beta^{(2)}_{1}(G)$ is
non-zero, it follows that $[p_{1}]$ is non-zero. Additionally, given the
faithfulness of $\tau$, we note that $\beta^{(2)}_{n}(G)$ vanishes for $n>1$,
leading to the vanishing of $p_{n}$.

###  4.1 Free product of finite cyclic groups

Consider the free product $G=\mathbb{Z}_{m}*\mathbb{Z}_{n}$ for $m\geq 2$ and
$n\geq 3$. These groups are known to be virtually free. The goal is to find
representatives for the K-class $[p_{1}]$ associated to these groups. As a
byproduct, we compute the delocalised $\ell^{2}$-Betti numbers of $G$.

######  Lemma 4.2.

Let $G=\mathbb{Z}_{m}*\mathbb{Z}_{n}$ for $m\geq 2$ and $n\geq 3$. The first
Laplacian $\Delta_{1}$ for $G$ is

$\Delta_{1}=d_{0}d_{0}^{*}+d_{1}^{*}d_{1}=\begin{bmatrix}2-s-s^{-1}+m\sum_{0\leq
i<m}s^{i}&(1-s^{-1})(1-t)\\\ (1-t^{-1})(1-s)&2-t-t^{2}+n\sum_{0\leq
j<n}t^{j}\end{bmatrix},$

where $s$ and $t$ are generators for $\mathbb{Z}_{m}$ and $\mathbb{Z}_{n}$,
respectively.

###### Proof.

In order to describe the Laplacian $\Delta_{1}$ we need a suitable cochain
complex. For that, we take the free resolution of the trivial
$\mathbb{Z}G$-module $\mathbb{Z}$ associated with the presentation complex of
$G$, which is

$\cdots\longrightarrow\mathbb{Z}G^{\oplus
2}\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\mathbb{Z}G^{\oplus
2}\stackrel{{\scriptstyle\delta_{0}}}{{\longrightarrow}}\mathbb{Z}G\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\mathbb{Z}\longrightarrow
0,$

where

$\displaystyle\epsilon(g)=1\quad g\in G$ $\displaystyle\delta_{0}(1,0)=1-s$
$\displaystyle\delta_{0}(0,1)=1-t$
$\displaystyle\delta_{1}(1,0)=\bigl{(}\sum_{0\leq i<m}s^{i},0\bigr{)}$
$\displaystyle\delta_{1}(0,1)=\bigl{(}0,\sum_{0\leq i<n}t^{j}\bigr{)}$

Applying the functor $\mathrm{Hom}_{\mathbb{Z}G}(\,\cdot\,,\ell^{2}(G))$ and
identifying the terms with direct sums of $\ell^{2}(G)$, yield

$\ell^{2}(G)\stackrel{{\scriptstyle
d_{0}}}{{\longrightarrow}}\ell^{2}(G)^{\oplus 2}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}\ell^{2}(G)^{\oplus 2}\longrightarrow\cdots,$

where the connecting maps can be identified with the operators

$\displaystyle d_{0}=\begin{bmatrix}1-s&1-t\end{bmatrix},\qquad
d_{1}=\begin{bmatrix}\sum_{0\leq i<m}s^{i}&0\\\ 0&\sum_{0\leq
j<n}t^{j}\end{bmatrix}.$

Taking the transpose and computing $\Delta_{1}=d_{0}d_{0}^{*}+d_{1}^{*}d_{1}$,
we obtain the desired operator.

The concrete $\Delta_{1}$ from Lemma 4.2 is an essential ingredient to the
subsequent main theorem.

######  Theorem 4.3.

Let $G=\mathbb{Z}_{m}\ast\mathbb{Z}_{n}$ with $m\geq 2$ and $n\geq 3$. The
K-class of the higher Kazhdan projection $p_{1}$ in
$\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$ can be represented by

$[p_{1}]=[1]-\left[\frac{1}{m}\sum_{0\leq
i<m}{s^{i}}\right]-\left[\frac{1}{n}\sum_{0\leq j<n}{t^{j}}\right].$

To describe the K-class of Kazhdan projection we will solve the equation
$\Delta_{1}x=0$, for $x\in\ell^{2}(G)^{\oplus 2}$. This requires some
preparatory lemmas. Before that we fix some notation. Notation. Let $p$ and
$q$ be the averaging projections associated with the trivial representations
of $\mathbb{Z}_{m}$ and $\mathbb{Z}_{n}$, respectively, i.e.

(4.1) $p=\frac{1}{m}{\sum_{0\leq i<m}{s^{i}}}\quad\text{and}\quad
q=\frac{1}{n}{\sum_{0\leq j<n}{t^{j}}}.$

Moreover, let

$C=\begin{bmatrix}p&0\\\ 0&q\end{bmatrix}$

be the projection associated with $p$ and $q$.
Further, for a Hilbert space $H$ we denote by $P_{H}$ its associated
projection. The following lemma characterises the $l^{2}$-solutions of
$\Delta_{1}x=0$.

######  Lemma 4.4.

Let $G=\mathbb{Z}_{m}\ast\mathbb{Z}_{n}$ with $m\geq 2$ and $n\geq 3$. Let
$x\in\ell^{2}(G)^{\oplus 2}$. Then $\Delta_{1}x=0$ if and only if there are
$z\in\ell^{2}(G)^{\oplus 2}$ and $a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)$
such that

$x=(I-C)z\qquad\text{and}\qquad\begin{bmatrix}1-s&0\\\
0&{1-t}\end{bmatrix}z=\frac{1}{\sqrt{2}}\begin{bmatrix}a\\\ -a\end{bmatrix}.$

###### Proof.

The first Laplacian $\Delta_{1}$ explicitly described in 4.2 can be rewritten
as

$\Delta_{1}=\begin{bmatrix}m^{2}p&0\\\
0&n^{2}q\end{bmatrix}+\begin{bmatrix}{1-s^{-1}}&0\\\
0&{1-t^{-1}}\end{bmatrix}\begin{bmatrix}1&1\\\
1&1\end{bmatrix}\begin{bmatrix}{1-s}&0\\\ 0&{1-t}\end{bmatrix}.$

Suppose $x\in\ker\Delta_{1}$. By applying $C$ to $\Delta_{1}x=0$, we find that

$\begin{bmatrix}m^{2}&0\\\ 0&n^{2}\end{bmatrix}Cx=\begin{bmatrix}m^{2}p&0\\\
0&n^{2}q\end{bmatrix}x=0.$

It follows that $x\in\ker C=\mathrm{im}(1-C)$. Hence, there exists
$z\in\ell^{2}(G)^{\oplus 2}$ such that $x=(1-C)z$. Considering
$1-s=(1-p)(1-s)$ and $1-t=(1-q)(1-t)$, we have

(4.2) $\begin{bmatrix}{1-s^{-1}}&0\\\
0&{1-t^{-1}}\end{bmatrix}\begin{bmatrix}1&1\\\
1&1\end{bmatrix}\begin{bmatrix}{1-s}&0\\\ 0&1-t\end{bmatrix}z=0.$

The unitary $U=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix}$
satisfies

$U\begin{bmatrix}2&0\\\ 0&0\end{bmatrix}U^{*}=\begin{bmatrix}1&1\\\
1&1\end{bmatrix}.$

Thus the equation (4.2) can be rewritten as $A^{*}Az=0$, where
$A=\begin{bmatrix}\sqrt{2}&0\\\ 0&0\end{bmatrix}U^{*}\begin{bmatrix}1-s&0\\\
0&1-t\end{bmatrix}$. Furthermore, $A^{*}Az=0$ is equivalent to $Az=0$. Hence,
we only need to solve the following equation

$U^{*}\begin{bmatrix}{1-s}&0\\\ 0&{1-t}\end{bmatrix}\begin{bmatrix}z_{1}\\\
z_{2}\end{bmatrix}=\begin{bmatrix}0\\\ a\end{bmatrix},$

for some $a\in l^{2}(G)$. Applying $U$ to the left hand side, we infer that
$\Delta_{1}x=0$ implies that there are $z\in\ell^{2}(G)^{\oplus 2}$,
$x=(I-C)z$ and $a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)$ satisfying

$\begin{bmatrix}1-s&0\\\
0&{1-t}\end{bmatrix}z=\frac{1}{\sqrt{2}}\begin{bmatrix}a\\\ -a\end{bmatrix}.$

This proves the first part of the statement. The converse is evident through
straightforward computation.

For later reference we consider the following factorization of $1-p$ and
$1-q$.

(4.3) $1-p=\frac{1}{m}[(1-s)+\cdots+(1-s^{m-1})]=k(1-s)=(1-s)k,$

for some $k$. Similarly, there exists $l$ such that

(4.4) $1-q=l(1-t)=(1-t)l.$

######  Remark 4.5.

Consider the canonical inclusion
$\mathrm{M}_{n}(\mathrm{C}^{*}_{\text{red}}(G))\subset\mathcal{B}(\ell^{2}(G)^{\oplus
n})$ induced by the left regular representation of the group. From this
perspective, every projection
$r\in\mathrm{M}_{n}(\mathrm{C}^{*}_{\text{red}}(G))$ gives rise to an
orthogonal projection $R\in\mathcal{B}(\ell^{2}(G)^{\oplus n})$. We identify
$r$ with $R$ and shall write $[r\ell^{2}(G)^{\oplus n}]$ or
$[R\ell^{2}(G)^{\oplus n}]$ to denote the K-class of the projection from
$\ell^{2}(G)^{\oplus n}$ to $r\ell^{2}(G)^{\oplus n}$. We identify then
$[r\ell^{2}(G)^{\oplus n}]=[R\ell^{2}(G)^{\oplus n}]=[r]\in\mathrm{K}_{0}(A)$.

We proceed to find representatives for the K-theory class of the projection
associated to $\ker\Delta_{1}$.

######  Lemma 4.6.

Let $k$ and $l$ be as in (4.3) and (4.4). The $\ell^{2}$-kernel of
$\Delta_{1}$ is

$\ker\Delta_{1}=\left\\{\begin{bmatrix}ka\\\ -la\end{bmatrix}\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}.$

Moreover, the projection $P_{\ker\Delta_{1}}$ has the same $K$-class as the
projection $P_{H_{1}}$ onto the Hilbert subspace

(4.5) $H_{1}:=\left\\{\begin{bmatrix}a\\\ -a\end{bmatrix}\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}\subset\ell^{2}(G)\oplus\ell^{2}(G).$

In other words,

$[P_{\ker\Delta_{1}}]=[P_{H_{1}}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G)).$

###### Proof.

In view of (4.3), (4.4) and Lemma 4.4, we have that $\Delta_{1}x=0$ if and
only if $x$ satisfies

$x=\begin{bmatrix}1-p&0\\\ 0&1-q\end{bmatrix}z=\begin{bmatrix}k&0\\\
0&l\end{bmatrix}\begin{bmatrix}1-s&0\\\
0&1-t\end{bmatrix}z=\begin{bmatrix}k&0\\\
0&l\end{bmatrix}\begin{bmatrix}\frac{1}{\sqrt{2}}a\\\
-\frac{1}{\sqrt{2}}a\end{bmatrix}$

for $z=(I-C)x$ and some $a\in\ell^{2}(G)$ satisfying
$a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)$. In other words, we have

$\displaystyle\ker\Delta_{1}=$ $\displaystyle\left\\{\begin{bmatrix}1-p&0\\\
0&1-q\end{bmatrix}z\leavevmode\nobreak\ \leavevmode\nobreak\
\mid\leavevmode\nobreak\ \leavevmode\nobreak\ \begin{bmatrix}1-s&0\\\
0&1-t\end{bmatrix}z=\begin{bmatrix}\frac{1}{\sqrt{2}}a\\\
-\frac{1}{\sqrt{2}}a\end{bmatrix},a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\begin{bmatrix}\frac{ka}{\sqrt{2}}\\\
\frac{-la}{\sqrt{2}}\end{bmatrix}\leavevmode\nobreak\ \leavevmode\nobreak\
\mid\leavevmode\nobreak\ \leavevmode\nobreak\
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}.$

Due to the factorization in (4.3) and (4.4), $k$ and $1-s$ are inverse to each
other in $(1-p)\ell^{2}(G)$ and $l$ and $1-t$ are so in $(1-q)\ell^{2}(G)$.
With respect to the decomposition
$\ell^{2}(G)=p\ell^{2}(G)\oplus(1-p)\ell^{2}(G)$, and together with the fact
that $1-s=(1-p)(1-s)$ and $1-t=(1-q)(1-t)$ the operator $v_{1}:=p+(1-s)$ is
invertible with the inverse given by $v_{1}^{-1}:=1+k$. The same holds for the
operator $v_{2}:=q+(1-t)$. Its inverse given by $v_{2}^{-1}:=1+l$. Consider
the invertible operator $V:=\begin{bmatrix}v_{1}&0\\\ 0&v_{2}\end{bmatrix}$ in
$\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$. We then have

$\displaystyle V(\ker\Delta_{1})=\ker(V\Delta_{1}V^{-1})$ $\displaystyle=$
$\displaystyle\left\\{\begin{bmatrix}(1-p)a\\\ -(1-q)a\end{bmatrix}\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\begin{bmatrix}a\\\ -a\end{bmatrix}\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}=H_{1}.$

This proves the statement about the kernel of $\Delta_{1}$. From this we infer
that in particular

$P_{\ker\Delta_{1}}=V^{-1}P_{\ker(V\Delta_{1}V^{-1})}V=V^{-1}P_{H_{1}}V$

which in turn implies that as $K$-theory elements

$[P_{\ker\Delta_{1}}]=[P_{\ker(V\Delta_{1}V^{-1})}]=[P_{H_{1}}].$

This finishes the proof.

Before presenting our proof for Theorem 4.3 we need two more technical lemmas.
Recall the Hilbert space $H_{1}$ defined in (4.5). We consider four more
Hilbert subspaces of $\ell^{2}(G)\oplus\ell^{2}(G)$.

(4.6) $\displaystyle H_{2}:=\left\\{\begin{bmatrix}c\\\ -c\end{bmatrix}\mid
c\in\mathrm{im}p+\mathrm{im}q\right\\}\quad\tilde{H}_{2}:=\left\\{\begin{bmatrix}pw\\\
qv\end{bmatrix}\mid v,w\in\ell^{2}(G)\right\\}$ (4.7) $\displaystyle\quad
H_{3}:=\left\\{\begin{bmatrix}b\\\ b\end{bmatrix}\mid
b\in\ell^{2}(G)\right\\}\quad\tilde{H}_{3}:=\left\\{\begin{bmatrix}(1-p)u\\\
(1-q)u\end{bmatrix}\mid u\in\ell^{2}(G)\right\\}$

The next lemma provides two decompositions of $\ell^{2}(G)\oplus\ell^{2}(G)$.

######  Lemma 4.7.

There are orthogonal direct sums of Hilbert spaces

$\ell^{2}(G)\oplus\ell^{2}(G)=H_{1}\oplus H_{2}\oplus
H_{3}\quad\text{and}\quad\ell^{2}(G)\oplus\ell^{2}(G)=H_{1}\oplus\tilde{H}_{2}\oplus\tilde{H}_{3}.$

###### Proof.

The identity

$(\mathrm{im}(1-p)\cap\mathrm{im}(1-q))^{\perp}=\mathrm{im}(1-p)^{\perp}+\mathrm{im}(1-q)^{\perp}=\mathrm{im}(p)+\mathrm{im}(q),$

implies that

(4.8)
$\ell^{2}(G)=(\mathrm{im}(1-p)\cap\mathrm{im}(1-q))+\mathrm{im}(p)+\mathrm{im}(q).$

Applying this, it is then straightforward to verify that
$\ell^{2}(G)\oplus\ell^{2}(G)=H_{1}\oplus H_{2}\oplus H_{3}$. Now we turn to
the second decomposition. It is easy to check that $H_{1}$ and
$\tilde{H}_{2}$, $H_{1}$ and $H_{3}$ are orthogonal. Using
$\mathrm{im}(p)\cap\mathrm{im}(q)=\\{0\\}$, we infer that $\tilde{H}_{2}\cap
H_{3}=\\{0\\}$. Next we show that their sum is $\ell^{2}(G)\oplus\ell^{2}(G)$.
From (4.8) it follows that

(4.9) $\ell^{2}(G)=\\{2a+pw-qv\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q),w,v\in\ell^{2}(G)\\}.$

We have then the following equality of Hilbert spaces:

(4.10) $\displaystyle H_{1}\oplus\tilde{H}_{2}\oplus H_{3}=$
$\displaystyle\left\\{\begin{bmatrix}a+pw+b\\\ -a+qv+b\end{bmatrix}\mid
w,v,b\in\ell^{2}(G),a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}$ (4.11)
$\displaystyle=$ $\displaystyle\left\\{\begin{bmatrix}a+pw+b\\\ a+pw+b-(2a+pw-
qv)\end{bmatrix}\mid
w,v,b\in\ell^{2}(G),a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}$ (4.12)
$\displaystyle=$ $\displaystyle\left\\{\begin{bmatrix}x\\\
x-y\end{bmatrix}\mid x,y\in\ell^{2}(G)\right\\}$ (4.13) $\displaystyle\cong$
$\displaystyle\ell^{2}(G)\oplus\ell^{2}(G).$

Let us briefly explain how we obtain these. As $b\in\ell^{2}(G)$ is arbitrary,
it implies that the first coordinate $a+pw+b$ of
$H_{1}+H_{2}+H_{3}\subseteq\ell^{2}(G)\oplus\ell^{2}(G)$ can span the entire
$\ell^{2}(G)$. Additionally, by (4.9) the difference $2a+pw-qv$ of the two
coordinates of $H_{1}+H_{2}+H_{3}\subseteq\ell^{2}(G)\oplus\ell^{2}(G)$ can
also cover all of $\ell^{2}(G)$. Thus the third equality holds. We replace
$H_{3}$ in $H_{1}+\tilde{H}_{2}+H_{3}$ by

$\tilde{H}_{3}:=\left\\{\begin{bmatrix}(1-p)b\\\ (1-q)b\end{bmatrix}\mid
b\in\ell^{2}(G)\right\\}$

to obtain an orthogonal direct sum. It is straightforward to check that
$H_{1},\tilde{H}_{2},\tilde{H}_{3}$ are mutually orthogonal subspaces of
$\ell^{2}(G)\oplus\ell^{2}(G)$ and that their direct sum
$H_{1}\oplus\tilde{H}_{2}\oplus\tilde{H}_{3}$ is a subspace of
$\ell^{2}(G)\oplus\ell^{2}(G)$. By (4.10)-(4.4) every element of
$\ell^{2}(G)\oplus\ell^{2}(G)$ can be represented by

$\begin{bmatrix}a\\\ -a\end{bmatrix}+\begin{bmatrix}pw\\\
qv\end{bmatrix}+\begin{bmatrix}b\\\ b\end{bmatrix},$

for some $w,v,b\in\ell^{2}(G)$ and some
$a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)$. The latter is equal to

$\begin{bmatrix}a\\\ -a\end{bmatrix}+\begin{bmatrix}p(w+b)\\\
q(v+b)\end{bmatrix}+\begin{bmatrix}(1-p)b\\\ (1-q)b\end{bmatrix}\in
H_{1}\oplus H_{2}\oplus\tilde{H}_{3}.$

Hence $\ell^{2}(G)\oplus\ell^{2}(G)\subseteq
H_{1}\oplus\tilde{H}_{2}\oplus\tilde{H}_{3}$. This finishes the proof.

Let us remark that $P_{H_{3}}$ and $P_{\tilde{H}_{3}}$ belong to
$\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$. First observe that

$P_{H_{3}}=\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix}\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G)).$

Recall that by our assumption $\Delta_{1}$ has a spectral gap $(0,c]$ for some
$c\in\mathbb{R}_{+}$. Choose a continuous function on $\mathbb{R}$ such that
$f$ vanishes when $x\leq 0$ and equals $1$ when $x>\frac{c}{2}$. Then
$P_{\ker\Delta_{1}}=f(\Delta_{1})\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$.
The invertible matrix $V\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$
from the proof of Lemma 4.6 conjugates $P_{H_{1}}$ to $P_{\ker\Delta_{1}}$.
Hence $P_{H_{1}}\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$. Consider
$P_{\tilde{H}_{2}}=\begin{bmatrix}p&0\\\
0&q\end{bmatrix}\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$. By Lemma
4.7 then we have

$P_{\tilde{H}_{3}}=1-P_{\tilde{H}_{2}}-P_{H_{1}}\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G)).$

The next lemma identifies the K-classes of $P_{H_{3}}$ and $P_{\tilde{H}_{3}}$
in $K$-theory.

######  Lemma 4.8.

Let the Hilbert spaces $H_{3}$ and $\tilde{H}_{3}$ be as in (4.6). The
projections associated with them satisfy

$[P_{H_{3}}]=[P_{\tilde{H}_{3}}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G)).$

###### Proof.

First note that by Lemma 4.7, the projection
$P_{H_{2}}=1-(P_{H_{1}}+P_{H_{3}})$ belongs to
$\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$. So by previous arguments we
have that
$P_{H_{i}},P_{\tilde{H}_{j}}\in\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$
for $i=1,2,3$ and $j=1,2$. To show that the projections associated with
$H_{3}$ and $\tilde{H}_{3}$ have the same K-class, we construct an invertible
operator in $\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$. Consider the
following bijective continuous linear maps associated with $H_{2}$ and
$\tilde{H}_{2}$ as well as $H_{3}$ and $\tilde{H_{3}}$.

$\displaystyle u:H_{2}$
$\displaystyle\rightarrow\tilde{H}_{2}\quad\begin{bmatrix}pw\\\
qv\end{bmatrix}\mapsto\begin{bmatrix}pw+qv\\\ -pw-qv\end{bmatrix}$
$\displaystyle v:H_{3}$
$\displaystyle\rightarrow\tilde{H}_{3}\quad\begin{bmatrix}x\\\
x\end{bmatrix}\mapsto\begin{bmatrix}(1-p)x\\\ (1-q)x\end{bmatrix}$

Note that injectivity is due to the fact that
$\mathrm{im}(p)\cap\mathrm{im}(q)=\\{0\\}$. Together with the fact that
$H_{i},\tilde{H}_{i}$ are Hilbert spaces, the algebraic inverse of $u,v$ are
also bounded. Employing these operators we construct an invertible operator
$U$ in $\mathcal{B}(\ell^{2}(G)^{\oplus 2})$.

$U=\begin{bmatrix}1&0&0\\\ 0&u&0\\\ 0&0&v\end{bmatrix}:H_{1}\oplus H_{2}\oplus
H_{3}\rightarrow H_{1}\oplus\tilde{H}_{2}\oplus\tilde{H}_{3}.$

The operator $U$ belongs to $\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$,
since we can view it as

$U=P_{H_{1}}+P_{\tilde{H}_{2}}\begin{bmatrix}1&1\\\
-1&-1\end{bmatrix}P_{H_{2}}+P_{\tilde{H}_{3}}\begin{bmatrix}1-p&0\\\
0&1-q\end{bmatrix}P_{H_{3}}.$

Since $\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$ is a
$\mathrm{C}^{*}$-algebra, it is closed under holomorphic functional calculus,
which is equivalent to being inverse closed. Hence the inverse $U^{-1}$ of $U$
belongs to $\mathrm{M}_{2}(\mathrm{C}^{*}_{\text{red}}(G))$ as well.
Therefore, in $K$-theory we have

$[P_{\tilde{H}_{3}}]=[U^{-1}P_{\tilde{H}_{3}}U]=[P_{H_{3}}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G)).$

Now we are ready to present our proof for Theorem 4.3.

###### Proof of Theorem 4.3.

We know by Lemma 4.1 that $p_{1}$ exists. Lemma 4.6 implies that
$[p_{1}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$ is represented by
the projection from $\ell^{2}(G)^{\oplus 2}$ onto $H_{1}$. In other words, we
have

$[p_{1}]=[P_{H_{1}}]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G)).$

We infer from Lemma 4.7 that

$[H_{1}]=[H\oplus H]-[\tilde{H}_{3}]-[pH]-[qH],$

which is by Remark 4.5 equal to

$[P_{H_{1}}]=2[1]-[P_{\tilde{H}_{3}}]-[p]-[q]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G)).$

Note that $[P_{\tilde{H}_{3}}]=[1]$. Putting all these together we conclude
that

$[p_{1}]=2[1]-[1]-[p]-[q]=[1]-[p]-[q]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G)).$

More concretely, we have

$[p_{1}]=[1]-\left[\frac{1}{m}\sum_{0\leq
i<m}{s^{i}}\right]-\left[\frac{1}{n}\sum_{0\leq j<n}{t^{j}}\right].$

When $m=2$ and $n=3$ the above proof can be slightly modified so that
$\ker\Delta_{1}$ can be expressed more concretely. We discuss this in the
following example.

######  Example 4.9.

Consider the group
$\mathrm{PSL}(2,\mathbb{Z})=\mathbb{Z}_{2}\ast\mathbb{Z}_{3}$. The K-class of
the higher Kazhdan projection $p_{1}$ can be described as

$[p_{1}]=[1]-\left[\frac{1+s}{2}\right]-\left[\frac{1+t+t^{2}}{3}\right]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(\mathrm{PSL}(2,\mathbb{Z}))).$

###### Proof.

Let $s,t$ be generators of order 2 and 3, respectively. By Lemma 4.2, the
Laplacian $\Delta_{1}$ can be expressed as

$\Delta_{1}=d_{0}d_{0}^{*}+d_{1}^{*}d_{1}=\begin{bmatrix}4&(1-s)(1-t^{2})\\\
(1-t)(1-s)&5+2t+2t^{2}\end{bmatrix}.$

As before set $p=\frac{1+s}{2}$ and $q=\frac{1+t+t^{2}}{3}$. Similar to (4.3)
and (4.4), we write

$1-p=\frac{1-s}{2}\frac{1-s}{2},\quad
1-q=\frac{1-t}{\sqrt{3}}\frac{1-t^{2}}{\sqrt{3}}.$

View $\Delta_{1}$ as

$\Delta_{1}=\begin{bmatrix}4p&0\\\
0&9q\end{bmatrix}+\begin{bmatrix}\frac{1-s}{2}&0\\\
0&\frac{1-t}{\sqrt{3}}\end{bmatrix}\begin{bmatrix}4&2\sqrt{3}\\\
2\sqrt{3}&3\end{bmatrix}\begin{bmatrix}\frac{1-s}{2}&0\\\
0&\frac{1-t^{2}}{\sqrt{3}}\end{bmatrix}.$

Following the argument as in Lemma 4.4 and Lemma 4.6 we have

$\displaystyle\ker\Delta_{1}=$ $\displaystyle\left\\{\begin{bmatrix}1-p&0\\\
0&1-q\end{bmatrix}z\mid\begin{bmatrix}\frac{1-s}{2}&0\\\
0&\frac{1-t}{\sqrt{3}}\end{bmatrix}z=\begin{bmatrix}-\frac{\sqrt{3}}{\sqrt{7}}a\\\
\frac{2}{\sqrt{7}}a\end{bmatrix}a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\begin{bmatrix}\frac{1-s}{2}(-\frac{\sqrt{3}}{\sqrt{7}}a)\\\
\frac{1-t^{2}}{\sqrt{3}}\frac{2}{\sqrt{7}}a\end{bmatrix}\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}.$

We infer that $[P_{\ker\Delta_{1}}]$ has the same $K$-class as the projection
onto

$H_{1}=\left\\{\begin{bmatrix}(1-p)a\\\ -(1-q)a\end{bmatrix}\mid
a\in\mathrm{im}(1-p)\cap\mathrm{im}(1-q)\right\\}.$

Therefore, we have

$[p_{1}]=[1]-[p]-[q]\in\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(\mathrm{PSL}(2,\mathbb{Z}))).$

As an immediate consequence of Theorem 4.3 we have the following result.

######  Corollary 4.10.

The delocalised $\ell^{2}$-Betti numbers for
$G=\mathbb{Z}_{m}\ast\mathbb{Z}_{n}$ are

$\beta^{(2)}_{1,\langle{g}\rangle}(G)=\begin{cases}1-\frac{1}{m}-\frac{1}{n}&\qquad
g=e\\\ -\frac{|\langle{g}\rangle|}{m}&\qquad
g\in\mathbb{Z}_{m}\setminus\\{e\\}\\\ -\frac{|\langle{g}\rangle|}{n}&\qquad
g\in\mathbb{Z}_{n}\setminus\\{e\\}\\\
0\qquad&\qquad\text{otherwise}\end{cases}$

and $\beta^{(2)}_{k,\langle{g}\rangle}(G)=0$ for $k\neq 1$ and $g\in G$.

###### Proof.

Due to hyperbolicity of $G$, the delocalised traces are well-defined on
$\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(G))$. The pairing between these
traces and $[p_{1}]$ from Theorem 4.3 implies the desired result for
delocalised $\ell^{2}$-Betti numbers in degree 1. In all other degrees this
invariant vanishes. This is because by Lemma 4.1, we have $p_{k}=0$ for $k\neq
1$.

In particular, when $m=2$ and $n=3$ we obtain the following result.

######  Example 4.11.

The delocalised $\ell^{2}$-Betti numbers for $G=\mathrm{PSL}(2,\mathbb{Z})$
are

$\beta^{(2)}_{1,\langle{g}\rangle}(G)=\begin{cases}1/6&\qquad g=e\\\
-1/2&\qquad g=s\\\ -1/3&\qquad g=t\\\ -1/3&\qquad g=t^{2}\\\
0&\qquad\text{otherwise}\end{cases}$

and $\beta^{(2)}_{k,\langle{g}\rangle}(G)=0$ for $k\neq 1$ and $g\in G$.

###  4.2 Certain product groups

Consider a product of free groups with a finite group. We show that higher
Kazhdan projections for such a group are not a multiple of the identity in any
fixed degree. We further describe their K-classes. We recall from [LNP20,
Example 1.5.1] an argument showing that the K-class of $p_{1}$ in
$\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2}))$ for the free
group $\mathbb{F}_{2}$ is $[1]$. To view $p_{1}$ as a member of a matrix
algebra over $\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2})$, a spectral gap for
$\Delta_{1}$ is required. For this we appeal to Lemma 2.2. In fact, the
cohomology group $\mathrm{H}^{1}(\mathbb{F}_{2},\ell^{2}(\mathbb{F}_{2}))$ is
reduced because $\mathbb{F}_{2}$ is non-amenable, and
$\mathrm{H}^{2}(\mathbb{F}_{2},\ell^{2}(\mathbb{F}_{2}))$ is reduced (even
vanishes) because the classifying space $\mathrm{B}\mathbb{F}_{2}$ has a one-
dimensional model (e.g. its presentation complex). Hence $\Delta_{1}$ has a
spectral gap. Now since

$\mathrm{K}_{0}(\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2}))=\langle{[1]}\rangle\cong\mathbb{Z},$

we conclude that $[p_{1}]=k[1]$, for some $k\in\mathbb{Z}$. Equation (2.1)
together with $\beta^{(2)}_{1}(\mathbb{F}_{2})=1$ imply that

$k=\tau([p_{1}])=1,$

hence $[p_{1}]=[1]$. It follows that $p_{1}$ is even Murray-von Neumann
equivalent to $1$. See [DHR97]

######  Lemma 4.12.

Let $G$ be a finite product of groups $G=G_{1}\times\cdots\times G_{k}$.
Assume that for $1\leq j\leq k$ there is a Hilbert chain complex such that the
associated Laplacians $\Delta_{i}^{G_{j}}$ has spectral gap for all
$i\in\mathbb{N}$. Then the Laplacians $\Delta_{i}^{G}$ associated to the
product complex of $G$ have spectral gap for all $i\in\mathbb{N}$.

###### Proof.

The result follows by induction once we prove it for the product of two groups
$G=G_{1}\times G_{2}$. Let $(C^{j,i})_{i\in\mathbb{N}}$ be Hilbert cochain
complexes whose associated Laplacians $(\Delta_{i}^{G_{j}})_{i}$ have spectral
gap. Then the coboundary maps of the associated product complex
$\bigoplus_{r+s=i}C^{1,r}\hat{\otimes}C^{2,s}$ are given by

$d^{i}=\bigoplus_{r+s=i}d^{r}\otimes I+(-1)^{s}I\otimes d^{s},$

and in turn the Laplacian $\Delta^{G}_{i}$ is given by

$\Delta^{G}_{i}=\bigoplus_{r+s=i}\Delta^{G_{1}}_{r}\otimes
I+I\otimes\Delta^{G_{2}}_{s}.$

By construction a Laplacian is a positive operator. Moreover, by our
assumption, $\Delta_{j}^{G_{i}}$’s have spectral gap for all $i\in\mathbb{N}$.
Therefore, for all $i\in\mathbb{N}$ the Laplacian $\Delta_{i}^{G}$ is a sum of
commuting positive operators with spectral gap and hence has spectral gap.

######  Theorem 4.13.

Let $F$ be a finite group, let $n\in\mathbb{N}$ and consider the product
$G=\mathbb{F}_{2}\times\dotsc\times\mathbb{F}_{2}\times F$ of $n$ factors of
$\mathbb{F}_{2}$ with $F$. Then the K-class of the $n$-th higher Kazhdan
projection $p_{n}$ of $G$ satisfies

$[p_{n}]=\left[\frac{1}{|F|}{\sum_{g\in F}{g}}\right].$

###### Proof.

Recall that the product resolution is calculating the cohomology of a product
of groups. We prove the following statement by induction on the number $n$ of
factors of $\mathbb{F}_{2}$. It immediately implies the theorem. For all
$n\in\mathbb{N}$, the Laplacian associated with the product resolution of $n$
factors of $\mathbb{F}_{2}$ with $F$ has spectral gap in all degrees and the
$n$-th Kazhdan projection is Murry-von Neumann equivalent in
$\mathrm{M}_{\infty}(\mathrm{C}^{*}_{\text{red}}(G))=\mathrm{M_{\infty}}(\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2})\otimes\cdots\otimes\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2})\otimes\mathrm{C}^{*}_{\text{red}}(F))$
to

$1\otimes\cdots\otimes 1\otimes\frac{1}{|F|}{\sum_{g\in F}{g}}.$

The claim about a spectral gap is the content of Lemma 4.12. We prove the
claimed Murray-von Neumann equivalence. The base case $n=0$ is trivial by a
known calculation. Assume that the statement holds for $n$ factors of
$\mathbb{F}_{2}$, we show it for $n+1$ factors. Consider the product
decomposition

$G=\mathbb{F}_{2}\times(\underbrace{\mathbb{F}_{2}\times\dotsc\times\mathbb{F}_{2}}_{n\text{
times}}\times F):=H\times K.$

Considering the iterated product resolution, we have

$C^{n+1}(H\times K,\ell^{2}(H\times
K))\cong\bigoplus_{r+s=n+1}C^{r}(H,\ell^{2}(H))\otimes C^{s}(K,\ell^{2}(K)).$

For $r\geq 2$, $C^{r}(H,\ell^{2}(H))$ is zero, so the corresponding term in
the direct sum vanishes. Moreover, when $s\geq n+1$, the same happens to
$C^{s}(K,\ell^{2}(K))$. Therefore, we find that

$C^{n+1}(H\times K,\ell^{2}(H\times K))\cong C^{1}(H,\ell^{2}(H))\otimes
C^{n}(K,\ell^{2}(K))$

and the associated Laplacian is

$\Delta^{G}_{n+1}=\Delta_{1}^{H}\otimes I+I\otimes\Delta_{n}^{K}.$

The kernel of this sum of positive commuting operators is equal to the
intersection of the kernels of $\Delta_{1}^{H}\otimes I$ and
$I\otimes\Delta_{n}^{K}$. So the kernel projection is
$p_{n+1}^{G}=(p_{1}^{H}\otimes 1)\cdot(1\otimes p_{n}^{K})=p_{1}^{H}\otimes
p_{n}^{K}$. The induction hypothesis says that there is a partial isometry $v$
whose support projection is $p_{n}^{K}$ and whose range projection is
$1\otimes\cdots\otimes 1\otimes\frac{1}{|F|}{\sum_{g\in F}{g}}$ with $n$
factors of the unit $1\in\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2})$. Since
$p_{1}^{H}$ is Murray-von Neumann equivalent to $1$, there is a partial
isometry
$u\in\mathrm{M}_{\infty}(\mathrm{C}^{*}_{\text{red}}(\mathbb{F}_{2}))$ whose
support projection is $p_{1}^{H}$ and whose range projection is $1$.
Considering the partial isometry $u\otimes v$, we have that

$p_{n+1}^{G}\stackrel{{\scriptstyle
MvN}}{{\sim}}\underbrace{1\otimes\cdots\otimes
1}_{n+1\,{\text{times}}}\otimes\frac{1}{|F|}{\sum_{f\in F}{f}}.$

This finishes the proof.

## References

  * [BHS10] Paul Baum, Nigel Higson, and Thomas Schick.  A geometric description of equivariant $K$-homology for proper actions.  In Quanta of maths, volume 11 of Clay Math. Proc., pages 1–22. Amer. Math. Soc., Providence, RI, 2010.
  * [BN15] Uri Bader and Piotr W. Nowak.  Cohomology of deformations.  J. Topol. Anal., 7(1):81–104, 2015.
  * [BN20] Uri Bader and Piotr W. Nowak.  Group algebra criteria for vanishing of cohomology.  J. Funct. Anal., 279(11), 2020.
  * [CG86] Jeff Cheeger and Mikhael Gromov.  $L_{2}$-cohomology and group cohomology.  Topology, 25:189–215, 1986.
  * [Dav80] E. Brian Davies.  One-parameter semigroups, volume 15 of Lond. Math. Soc. Monogr. Academic Press, London, 1980.
  * [Del77] Patrick Delorme.  1-cohomologie des représentations unitaires des groupes de Lie semi-simples et resolubles. Produits tensoriels continus de représentations.  Bull. Soc. Math. France, 105(3):281–336, 1977.
  * [DHR97] Ken Dykema, Uffe Haagerup, and Mikael Rørdam.  The stable rank of some free product $C^{*}$-algebras.  Duke Math. J., 90(1):95–121, 1997.
  * [Laf02] Vincent Lafforgue.  $K$-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes.  Invent. Math., 149(1):1–95, 2002.
  * [LNP20] Kang Li, Piotr W. Nowak, and Sanaz Pooya.  Higher Kazhdan projections, $\ell^{2}$-Betti numbers and Baum-Connes conjectures.  https://arxiv.org/abs/2006.09317, J. Noncommutative Geom., to appear, 2020.
  * [Lot99] John Lott.  Delocalized $L^{2}$-invariants.  J. Funct. Anal., 169(1):1–31, 1999.
  * [Pus10] Michael Puschnigg.  New holomorphically closed subalgebras of $C^{*}$-algebras of hyperbolic groups.  Geom. Funct. Anal., 20(1):243–259, 2010.
  * [WW16] Bai-Ling Wang and Hang Wang.  Localized index and $L^{2}$-Lefschetz fixed-point formula for orbifolds.  J. Differential Geom., 102(2):285–349, 2016.

Sanaz Pooya
Institute of Mathematics
University of Potsdam
14476 Potsdam, Germany
<EMAIL_ADDRESS>

Hang Wang
Research Center of Operator Algebras
East China Normal University
Shanghai 200241, China
<EMAIL_ADDRESS>