#  Enhanced superconductivity and moderate spin fluctuations suppressed at low
energies in heavily electron-doped La1111-based superconductor

T. Kouchi<EMAIL_ADDRESS>S. Nishioka  K. Suzuki  M. Yashima
H. Mukuda<EMAIL_ADDRESS>Graduate School of Engineering Science,
Osaka University, Osaka 560-8531, Japan  T. Kawashima  H. Tsuji  K. Kuroki  S.
Miyasaka  S. Tajima  Graduate School of Science, Osaka University, Osaka
560-0043, Japan

###### Abstract

To elucidate the origin of the re-enhanced high-$T_{c}$ phase in the heavily
electron-doped Fe pnictides, systematic 75As NMR studies are performed on
heavily electron-doped LaFe$Pn$O0.75H0.25 by controlling the pnictogen height
($h_{Pn}$) from the Fe plane through the substitution at $Pn$(= As) site with
Sb or P. The measurements of nuclear spin relaxation rate (1/$T_{1}$) and
Knight shift ($K$) reveal that the moderate spin fluctuations at high
temperatures are suppressed toward low temperatures. Such characteristic spin
fluctuations with gap-like feature at low-energies are more enlarged in higher
$T_{c}$ compounds with higher $h_{Pn}$, while those are totally suppressed in
non-superconducting compounds with lower $h_{Pn}$. This implies that the
contribution of the finite-energy part in the spin fluctuation spectrum is
crucial for enhancing $T_{c}$ in the heavily electron-doped regime. This is in
contrast to many cases of typical Fe-based compounds with hole and electron
Fermi surfaces of similar sizes, where the spin fluctuations at low energies
develop significantly at low temperatures. The features in the heavily
electron-doped states are discussed in relation with the characteristics of
the faint hole Fermi surface derived from the $d_{xy}$ orbital that rises when
$h_{Pn}$ is high, together with the enhanced electron correlation effects.

###### pacs:

74.70.Xa, 74.25.Ha, 76.60.-k

##  I Introduction

Superconductivity (SC) in typical iron pnictides (Fe$Pn$) LaFeAsO1-yFy
($T_{c}$= 26 K) [1] emerges in the vicinity of a stripe-type antiferromagnetic
(AFM) order accompanied by a structural phase transition from tetragonal to
orthorhombic (Ort.) phase. The parent electronic states are composed of hole
and electron Fermi pockets in similar sizes. Remarkable high-$T_{c}$ states
have been discovered in the heavily electron-doped Fe-based compounds, such as
single-layer FeSe ($T_{c}\geq$65 K) [2, 3], intercalated FeSe systems
($T_{c}$=30-50 K) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], and
LaFeAsO1-y(F/H)y ($T_{c}$$\sim$36 K) [19, 20]. These high-$T_{c}$ states
appear in the characteristic electronic states composed of large electron
Fermi surface (FS) and no hole FS or the faint hole FS. Because neither the
magnetic nor the orbital order phases has been reported universally in the
vicinity of their SC phases, the indispensable factor for enhancing $T_{c}$ is
still unclear. Toward coherent understanding of many Fe-based SCs, it is
important to reveal universalities and/or diversities of the SC mechanisms
over a wide doping region.

Here, we focus on the reemergent high-$T_{c}$ phase of heavily electron-doped
1111-compounds LaFe$Pn$O1-yHy ($Pn$ = As1-xSbx,
As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$) [19, 21, 22, 23]. As shown in Fig.
1(a), in this series the electron doping level can be broadly controlled by
the content $y$ from the lightly doped SC phase (SC1) to the re-enhanced
higher SC phase (SC3) in a heavily electron-doped regime [19, 21, 22, 23]. The
$T_{c}$ within the SC3 phase is enhanced by substitution of the As site with
Sb as shown in Fig. 1(b), which increases the pnictogen height ($h_{Pn}$) from
the Fe plane [22], whereas the P substitution at the As site decreases
$h_{Pn}$ and significantly reduces $T_{c}$. Theoretically, it is expected that
the variation in $h_{Pn}$ has a significant influence on the energy level of
the faint hole FS mainly from the Fe-$3d_{xy}$ orbital [24] as shown in Fig.
1(c), which increases (decreases) when $h_{Pn}$ becomes high (low), in
addition to the electron correlation effects that are more enhanced (degraded)
[25, 26]. Therefore, this is a unique system to elucidate the roles of faint
hole FS in the vicinity of Fermi level ($E_{\rm F}$) in the viewpoint of the
relationship between the normal electronic state and the SC state. In heavily
electron-doped SC phases such as LaFeAsO1-y(F/H)y [20, 27, 28, 29] and
intercalated FeSe [30, 31, 32, 33, 34, 35], spin fluctuations critically
enhanced at low energies have not been clearly observed in previous NMR
measurements, which is in contrast to the cases of the lightly electron-doped
SC states (SC1 [36] and SC2 [37, 38]), where the spin fluctuations at low
energies are significantly enhanced toward low temperatures. NMR study is
advantageous to elucidate the evolution of the electronic states by means of a
common 75As nuclear probe within a same family compound from the lightly
electron-doped states (SC1 and SC2) to the heavily electron-doped state (SC3).

In this study, we report systematic 75As NMR studies on LaFe$Pn$(O0.75H0.25)
for 0$\leq x\leq$0.4 ($Pn$=As1-xSbx) and 0$\leq x^{\prime}\leq$0.4
($Pn$=As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$), revealing that characteristic
spin fluctuations suppressed at low energies appear explicitly in the Sb-
substituted high-$T_{c}$ compounds, whereas those are totally suppressed in
P-substituted non-SC compounds. These results suggest that the contribution of
the finite energy part in the spin fluctuation spectrum may be rather
important for enhancing $T_{c}$ in heavily electron-doped states (SC3), which
differs from that of the typical lightly doped Fe-pnictides (SC1/SC2).

Samples |  $Pn$ site |  $T_{c}$ (K) |  $h_{Pn}$(Å) |  75$\nu_{\rm Q}$ (MHz)
---|---|---|---|---
$x$=0.4 | As0.6Sb0.4 | 32.1 | $1.371$ | 10.2
$x$=0.3 | As0.7Sb0.3 | 33.1 | 1.368 | 10.3
$x$=0.2 | As0.8Sb0.2 | 32.1 | 1.361 | 10.2
$x$=0 | As | 15.3 | 1.326 | 10.1
$x^{\prime}$=0.2 | As0.8P0.2 | 0 | 1.270 | 9.7
$x^{\prime}$=0.4 | As0.6P0.4 | 0 | 1.258 | 9.0

Figure 1: (Color online) (a) Phase diagram of LaFe$Pn$O1-y(F/H)y
($Pn$=As1-xSbx and As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$) for $x=0$ [19,
39], 0$\leq$$y$$\leq$0.1 [1, 40, 41, 42, 43, 44, 45, 46, 47, 21] and $y$=0.25
[21, 22]. (b) The contents ($x$ and $x^{\prime}$) dependence of $T_{c}$ for
$y$=0.25 investigated in this study [22]. (c) Schematics of possible band
structure made by hole and electron Fermi surfaces around SC3. The theoretical
study suggested that the $d_{xy}$ hole FSs around ($\pi,\pi$) increase
(reduces) by Sb substitution (P substitution) [24]. The table summarizes the
properties of LaFe$Pn$(O0.75H0.25) used in this study. The values of $T_{c}$
and $h_{Pn}$ are cited from the previous reports [40, 21, 22]. The $\nu_{\rm
Q}$’s at the 75As site are obtained in this work.

##  II experimental

Polycrystalline samples of LaFe$Pn$O0.75H0.25 with nominal contents at $0\leq
x\leq 0.4$ ($Pn$=As1-xSbx) and $0\leq x^{\prime}\leq 0.4$
($Pn$=As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$) were synthesized using a solid
state reaction method as described elsewhere [22]. Powder x-ray diffraction
measurements indicate that the lattice parameters exhibit a monotonic
variation with $x$($x^{\prime}$) [22]. The bulk $T_{c}$ values were determined
from an onset of zero-resistivity and diamagnetic response in dc
susceptibility measurement [22, 23]. As shown in Fig. 1(b), the $T_{c}$ in the
SC3 phase at $y$=0.25 increases to $T_{c}\sim 33$ K by Sb substitution ($x\sim
0.4$), whereas the $T_{c}$ decreases to zero by P substitution at
$x^{\prime}>0.2$ [22]. The 75As NMR measurements were performed on coarse-
powder samples of LaFe$Pn$O0.75H0.25 with $x=0,0.2,0.3,0.4$ ($Pn$=As1-xSbx)
and $x^{\prime}=0.2,0.4$ ($Pn$=As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$) for
the fixed electron-doping level at $y=0.25$. The Knight shift ($K$) was
evaluated by a narrow central peak of the 75As NMR spectra for the field
direction of the $H_{0}\perp c$ axis. The nuclear spin-lattice relaxation rate
($1/T_{1}$) was measured at $H_{0}$$\sim 8$ T, which was determined by fitting
a recovery curve for 75As nuclear magnetization to a multiple exponential
function $m(t)=0.1\exp(-t/T_{1})+0.9\exp(-6t/T_{1})$.

Figure 2: (Color online) (a) 75As NMR spectra at $T=40$ K for all the samples,
obtained at a fixed frequency 58.634 MHz. (b) $T$ dependence of 75$K$ in
LaFe$Pn$O0.75H0.25 for $0\leq x\leq 0.4$ ($Pn$=As1-xSbx) and $0\leq
x^{\prime}\leq 0.4$ ($Pn$=As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$). The 75$K$
decreases monotonically upon cooling in all the samples. (c) $T$ dependences
of ${}^{75}(1/T_{1}T)$ for each sample are more enlarged in Sb-substituted
compounds, although the $T$ dependence of 75$K$ is almost comparable.

##  III Results and Discussion

The 75As NMR spectra of well-defined powder pattern ($I$=3/2) were obtained
for all the samples, as shown in Fig. 2(a). According to the second-order
perturbation theory for the nuclear Hamiltonian with $H_{0}\perp c$ [48, 49],
the NMR shifts consist of the Knight shift ($K$) and the second-order
quadrupole shift, as expressed by

$\left(\frac{f_{0}-\gamma_{\rm N}H_{\rm res}}{\gamma_{\rm N}H_{\rm
res}}\right)=K+\frac{3\nu^{2}_{\rm Q}}{16(1+K)}\frac{1}{(\gamma_{\rm N}H_{\rm
res})^{2}},$ (1)

where $\gamma_{\rm N}$ is a nuclear gyromagnetic ratio, $H_{\rm res}$ is a
resonance field, and $\nu_{\rm Q}$ is a nuclear quadrupole frequency at the
75As site. Here, the electric field gradient asymmetry parameter ($\eta$) is
assumed to be close to zero for all the samples in this study, since it was
reported previously that $\eta\sim 0$ for $y<0.3$ in heavily electron-doped
LaFeAsO1-yFy [20]. To evaluate 75$K$ and ${}^{75}\nu_{\rm Q}$, the $H_{\rm
res}$ was measured as a function of the frequency $f_{0}$. The slope in the
plot of ($f_{0}-\gamma_{\rm N}H_{\rm res}$)/$\gamma_{\rm N}H_{\rm res}$
against ($\gamma_{\rm N}H_{\rm res}$)-2 gives an estimation of
${}^{75}\nu_{\rm Q}$ for each sample. As summarized in the table of Fig. 1,
the ${}^{75}\nu_{\rm Q}$ is roughly in proportion to the $h_{Pn}$. Note that
the change in ${}^{75}\nu_{\rm Q}$ is small for $x\geq 0.3$, which may be due
to the small change in $h_{Pn}$ [22]. Since the ${}^{75}\nu_{\rm Q}$ is
proportional to the electric field gradient at the 75As site, the result
ensures the success of monotonic variation of $h_{Pn}$ by the contents
$x$($x^{\prime}$) from the microscopic point of view.

Figure 3: (Color online) (a) Plot of 75(1/$T_{1}T$)0.5 vs. ${}^{75}K$ for
LaFe$Pn$O0.75H0.25 in normal states with an implicit parameter of $T$. It is
almost linear for $x^{\prime}$= 0.4, as shown by the broken line, which
enables us to estimate $K_{\rm chem}\sim$0.11%. (b)–(g) $T$ dependence of
${}^{75}(1/T_{1}T)$ and $K_{\rm s}^{2}$. The hatched regions correspond to the
component of (1/$T_{1}T$)AFM. (h) $T$ dependence of 75($1/T_{1}T$)AFM,
indicating that the AFMSFs appear at high temperature but these are suppressed
gradually toward low temperatures, which is more significant in $x\geq 0$ when
the $h_{Pn}$ is high.

Figures 2(b) and 2(c) show the temperature ($T$) dependence of $K$ and
${}^{75}(1/T_{1}T)$, respectively, for $0\leq x\leq 0.4$ ($Pn$=As1-xSbx) and
$0\leq x^{\prime}\leq 0.4$ ($Pn$=As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$). The
${}^{75}K$ decreases upon cooling for all the compounds, which is universal
for electron-doped Fe-based compounds. The $K$ comprises the spin part $K_{\rm
s}(T)$ and the $T$-independent chemical part $K_{\rm chem}$, as expressed by
$K(T)=K_{\rm s}(T)+K_{\rm chem}$. The $K_{\rm s}(T)$ is proportional to
uniform susceptibility $\chi(q$=$0)$ with the relation $K_{\rm s}(T)=A_{\rm
hf}(0)\chi(q$=$0)\propto A_{\rm hf}(0)N(E_{\rm F})$, where $A_{\rm hf}(0)$ is
the hyperfine coupling constant at $q$=$0$, and $N(E_{\rm F})$ is the density
of states (DOS) at the Fermi level ($E_{\rm F}$). In non correlated normal
metals, we expect the relation $(1/T_{1}T)\propto N(E_{\rm F})^{2}$, which
corresponds to the Korringa relation expressed as $(1/T_{1}T)^{0.5}\propto
K_{\rm s}$. To extract $K_{\rm chem}$, the ${}^{75}(1/T_{1}T)^{0.5}$ is
plotted against ${}^{75}K(T)$ for all the compounds, as shown in Fig. 3(a). As
for $x^{\prime}=0.4$, the plot shows almost linear relation (Korringa
relation) in the whole $T$ range, indicating that the contribution of the spin
fluctuations is negligibly small. It enables us to estimate $K_{\rm chem}$ to
be $\sim$0.11% for these compounds. On the other hand, in the SC compounds of
$x\geq 0$, the deviation from the linear relation is clearly seen, indicating
that the $(1/T_{1}T)$ includes additional contribution from the
antiferromagnetic spin fluctuations (AFMSFs) at finite wave vectors.

To extract the component derived from the AFMSFs, we assume that the
(1/$T_{1}T$) is decomposed as (1/$T_{1}T$) = (1/$T_{1}T$)AFM \+ (1/$T_{1}T$)0,
according to the previous studies [50, 37, 53, 51, 52]. The first term
(1/$T_{1}T$)AFM represents the component of AFMSFs enhanced at the finite wave
vectors (${\bm{q}}$) at low energies ($\omega\to 0$), which is described as

$\left(\frac{1}{T_{1}T}\right)_{\rm AFM}\propto\lim_{\omega\to
0}\sum_{\bm{q}}A_{\rm
hf}(\bm{q})^{2}\frac{\chi^{\prime\prime}(\bm{q},\omega)}{\omega},$ (2)

where $A_{\rm hf}(\bm{q})$ is the hyperfine-coupling constant at $\bm{q}$ and
$\chi^{\prime\prime}(\bm{q},\omega)$ is dynamical spin susceptibility at
finite wave vector $\bm{q}$ and energy $\omega$. The second term (1/$T_{1}T$)0
is the component related to $N(E_{\rm F})^{2}$, or $K_{\rm s}^{2}(T)$ [=
($K-K_{\rm chem})^{2}$]. Figures 3(b)–3(g) show the $T$ dependence of
75(1/$T_{1}T$) and $K_{\rm s}^{2}(T)$ for each sample. Since the $K_{s}(T)$
shows monotonous decreases upon cooling, the hatched regions in this figure
correspond to the component of $(1/T_{1}T)_{\rm AFM}$. Consequently, the $T$
dependence of ${}^{75}(1/T_{1}T)_{\rm AFM}$ is summarized in Fig. 3(h). We
note that in the high-$T_{c}$ compounds ($x\geq 0$) the spin fluctuations
develop at the high $T$ region moderately, but these are suppressed gradually
below $\sim$100 K. It indicates that even though the spin fluctuations are
dominant in these compounds, the gap like feature appears at low energies in
the spin fluctuation spectrum toward low temperatures.

Figure 4: (Color online) Contour plot of 75($1/T_{1}T$)AFM as functions of $T$
and $x$($x^{\prime}$) in LaFe$Pn$O1-y(F/H)y ($Pn$=As1-xSbx and
As${}_{1-x^{\prime}}$P${}_{x^{\prime}}$). The horizontal axis also corresponds
to the $h_{Pn}$(see the table in Fig. 1). The spin fluctuations exhibit a
moderate peak at high temperatures and a suppression toward low temperatures
when the $h_{Pn}$ is high, i.e., $x$ is large. This behavior of spin
fluctuations is more significant in $x\geq 0$ with higher $h_{Pn}$, whereas it
is largely suppressed in $x^{\prime}\geq 0.2$ with low $h_{Pn}$, suggesting a
possible relation with the reenhancement of $T_{c}$ in the heavily electron-
doped SC3 phase.

To reveal the relationship between the $h_{Pn}$, the characteristics of spin
fluctuations, and $T_{c}$, the contour plot of 75(1/$T$1$T$)AFM is shown in
Fig. 4, as functions of content $x$($x^{\prime}$) and $T$. The plot reveals
explicitly that the spin fluctuations enhanced at high temperatures are more
significant for the higher $T_{c}$ compounds with higher $h_{Pn}$ (larger
$x$), while those are largely suppressed in non-SC compounds with lower
$h_{Pn}$ ($x^{\prime}\geq 0.2$). It suggests that such characteristic spin
fluctuations suppressed at low energies are related with the enhancement of
$T_{c}$ in the heavily electron-doped SC3 phase. The suppression of low energy
spin fluctuations at low temperatures was also reported in the high-$T_{c}$
region of heavily electron-doped LaFeAsO1-yFy [20], which is similar to our
results. This feature differs from that of the SC1 and SC2 phases of lightly
electron-doped La1111, where the spin fluctuations at low energies are
significantly developed toward low temperatures, which has a relation with the
enhancement of $T_{c}$ [36, 37]. The significant difference of SC3 in the
heavily electron-doped regime can be attributed to the unique FS topology
dominated by the enlarged electron FS and faint hole pocket. The band
calculation points out that the energy level of the $d_{xy}$ orbital that
forms the faint hole pocket also increases when the $h_{Pn}$ increases [24],
which suggests a possible relation with unique spin fluctuations that appears
at $x>0$. In this FS topology, it is theoretically suggested that the finite
energy spin fluctuations give a pairing glue effectively in the heavily
electron-doped SC states [54, 55], even in the absence of well-nested FSs and
low energy spin fluctuations. Here we also note another important feature,
that is, stronger electron correlation effect derived from the higher $h_{Pn}$
[25, 26], which is anticipated in the SC3 phase as well. In the case of
heavily electron-doped Ba(Fe1-xCox)2As2 [50], both superconductivity and spin
fluctuations disappear at $x>0.15$, because $h_{Pn}$ becomes low when Fe is
replaced with Co [56]. Generally, in the unconventional SCs caused by strong
correlation effects, it has been discussed theoretically that the lack of the
low energy spin fluctuations that gives rise to a pair breaking may be rather
favorable [57, 58, 54, 59, 55, 60]. Thus, we suggest that one possible reason
for the enhanced $T_{c}$ in the heavily electron-doped SC3 phase is the finite
energy spin fluctuations optimized when the FSs are not nested, if the spin
fluctuations are a unique factor related to the SC mechanism.

Finally, we compare the feature of the SC3 phase with that of the lightly
electron-doped SC1 and SC2 phases. It should be noted that the P substitution
effect on $T_{c}$ of SC3 is quite different from that in the SC1 and SC2
phases. As shown in Fig. 1(a), the SC3 phase disappears drastically by P
substitution ($x^{\prime}>0.2$) while the SC1 phase is robust and the $T_{c}$
is enhanced again in the SC2 phase at further P substitution [43, 44, 46, 47].
The SC1 phase is dominated by the hole and electron FSs in similar sizes,
which are mainly derived from the $d_{xz}$/$d_{yz}$ orbitals, whereas the
composition of the $d_{xy}$ orbital is negligibly tiny since the $h_{Pn}$ is
low [24]. The nesting of hole and electron FSs derived mainly from
$d_{xz}$/$d_{yz}$ orbitals becomes better especially around the SC2/AFM2
region, which brings about the spin fluctuations at low energies developed
significantly toward low temperatures [53, 51, 61, 36, 37]. It plays an
indispensable role for enhancing $T_{c}$ in the SC1 and SC2 phases [53, 61,
51, 36, 37]. Here, in contrast to the SC3, the contribution of the $d_{xy}$
orbital is expected to be negligible since the energy level is far below
$E_{\rm F}$ owing to the low $h_{Pn}$. Consequently, although spin
fluctuations may be generally important for Cooper pairing in Fe-based
compounds over the broad doping region, we suggest that the finite energy
component of the spin fluctuations may play an important role for enhancing
the $T_{c}$ in the case of the heavily electron-doped Fe-based compounds. On
the other hand, in the SC3 region, it is possible that the other low energy
local fluctuations such as the orbital fluctuations could play some roles to
contribute to the superconductivity, since the structural transition with the
AFM3 phase appears in the vicinity of the SC3 phase. However, there is no
experimental results to detect the unknown fluctuations except for spin
fluctuations due to the lack of large single crystals. Further spectroscopic
experiments that reveal the possible roles of orbital degrees of freedom are
necessary for a general understanding of the SC mechanism in various Fe-based
compounds over wide doping regions.

##  IV Summary

In summary, the high-$T_{c}$ phase (SC3) re-enhanced in the heavily electron-
doping on LaFe$Pn$O0.75H0.25 was investigated by the common 75As NMR probe by
controlling the pnictogen height through the substitution at the $Pn$ site. We
revealed that the moderate spin fluctuations at high temperatures are
gradually suppressed toward low temperatures in the high-$T_{c}$ compounds
with high $h_{Pn}$, suggesting the gap like feature appears at low energy
regions in the spin fluctuation spectrum toward low temperatures. This
behavior is more significant when the $h_{Pn}$ is higher in the higher $T_{c}$
compounds, whereas it is more suppressed in the lower $T_{c}$ compounds with
lower $h_{Pn}$. The comparison with the theories on the FS topology suggest
that the key element may be the presence of the faint hole FS derived from the
$d_{xy}$ orbital and strong correlation effects, suggesting the importance of
the finite-energy AFMSFs in the heavily electron-doped states without the
nested FSs. A similar phenomenon was recently reported in intercalated FeSe
compounds [62] that was dominated by the large electron FS and faint hole FS
or no hole FS. We suggest that this type of AFMSFs may be a characteristic
feature, and one of the indispensable factors for enhancing $T_{c}$ in heavily
electron-doped SC states in iron based compounds. However, in these compounds,
the contribution of orbital degrees of freedom is still unclear. Further
general understanding of the SC mechanism in Fe-based compounds over a broad
doping region should be completed including the possible roles of orbital
degrees of freedom, that should be clarified by systematic spectroscopic
experiments as well as NMR probes in the future.

## Acknowledgements

T.K. is supported by a JSPS Fellowship (Grant No. 21J14053) and the Kato
Foundation for Promotion of Science (Grant No. KS-3227). This work was
supported by JSPS KAKENHI (Grants No. 16H04013 and No. 18K18734), the Murata
Science Foundation, the Mitsubishi Foundation, and the Tanigawa Fund.

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# Mixstyle based Domain Generalization for
Sound Event Detection with Heterogeneous Training Data

###### Abstract

This work explores domain generalization (DG) for sound event detection (SED),
advancing adaptability towards real-world scenarios. Our approach employs a
mean-teacher framework with domain generalization to integrate heterogeneous
training data, while preserving the SED model performance across the datasets.
Specifically, we first apply mixstyle to the frequency dimension to adapt the
mel-spectrograms from different domains. Next, we use the adaptive residual
normalization method to generalize features across multiple domains by
applying instance normalization in the frequency dimension. Lastly, we use the
sound event bounding boxes method for post-processing. Our approach integrates
features from bidirectional encoder representations from audio transformers
and a convolutional recurrent neural network. We evaluate the proposed
approach on DCASE 2024 Challenge Task 4 dataset, measuring polyphonic SED
score (PSDS) on the DESED dataset and macro-average pAUC on the MAESTRO
dataset. The results indicate that the proposed DG-based method improves both
PSDS and macro-average pAUC compared to the challenge baseline.

Index Terms—  sound event detection, semi-supervised learning, domain
generalization, mixstyle

##  1 Introduction

Sound event detection (SED) [1, 2] involves identifying and classifying sound
events from acoustic signals along with their timestamps across various
environments. In the recent years, deep learning models [3, 4, 5, 6, 7, 8]
have been witnessed success in SED research. However, these models require
large amount of strongly labelled data, which are costly and time-consuming to
obtain. To address these issues, weakly supervised or semi-supervised learning
techniques are used more frequently and also considered under the latest
edition of DCASE 2024 Challenge Task 4 [9]. The current edition utilizes a new
dataset namely, MAESTRO Real [10] that is softly labelled together with widely
used DESED [11] dataset containing strongly labelled data for encouraging SED
model development under heterogeneous training conditions. However, training a
robust SED model with heterogeneous training data is challenging due to the
fact that the labels may not be consistent across different datasets apart
from the domain mismatch due to various differences in the multiple datasets.

Deep neural networks (DNNs) often struggle to generalize to unseen domains due
to the domain mismatch mentioned above, leading to poor results in real-world
scenarios. Domain generalization (DG) [12] has become an essential research
topic across fields such as computer vision, audio processing, and natural
language processing. Techniques like adversarial gradients, feature-based
augmentation, and style transfer normalization have been used to address
device domain mismatch, such as in the DCASE challenge Task 1 [13]. Thus,
employing domain generalization for heterogeneous training data from different
domains for sound event detection remains underexplored. In previous editions
of DCASE Challenge Task 4 [14], researchers have explored solutions for domain
mismatch. For example, domain adaptation methods have been used to efficiently
exploit synthetic strong-labeled data by using domain classifiers, considering
the gap between synthetic and real audio data in the DESED dataset [11].

The DCASE 2024 Challenge Task 4 focuses on development of a single SED model
for detection of sound classes in DESED and MAESTRO datasets using their
training set collectively, in contrast to the previous edition in 2023 [5],
where separate SED model development for the two dataset was required. The
goal of the SED model is still to provide event classes along with their time
boundaries, even with multiple overlapping events. At the same time this task
emphasizes on leveraging training data with varying annotation granularity
(temporal resolution, soft/hard labels). Systems evaluated on the basis of
labels with different granularity can help to understand their behavior and
robustness for various applications. In addition, the target classes in
different datasets also differ, so sound labels present in one dataset might
not be annotated in another. Therefore, the developed SED system needs to
handle potentially missing target labels during training and perform without
knowing the origin of the audio clips at evaluation time.

In order to address the domain mismatch issue in DCASE 2024 Challenge Task 4,
we propose a novel mixstyle-based DG approach for SED with heterogeneous
training data. We first analyze the relationship between the domain and the
statistics of each feature dimension. Specifically, we mix styles of training
instances in the frequency dimension, resulting in novel domains being
synthesized implicitly. This increases the domain diversity of the source
domains and thereby enhances the generalizability of the trained model. In
addition, we use the adaptive residual normalization (AdaResNorm) method to
generalize features across multiple domains by applying instance normalization
in the frequency dimension. Further, we use the sound event bounding boxes
method for post-processing. With above methods, the developed SED system is
expected to perform more effectively even though the training data originates
from different sources. We also submitted our developed system based on the
proposed method for DCASE 2024 Challenge Task 4 participation [15].

##  2 Proposed Domain Generalization for SED

###  2.1 Analyzing domain mismatch in heterogeneous data

Domain mismatch in heterogeneous data is mostly due to the original sources of
data or how they were collected. In DCASE 2024 Task 4 Challenge, the audio
files in DESED were sampled from public sources, such as Freesound and
YouTube, while audio clips in MAESTRO was recorded in real-life scenarios.
Although all the sound classes in DESED and MAESTRO datasets are not same,
some events in DESED are mapped to similar classes in MAESTRO. For example, in
DESED, “speech” is a super-class for “people talking,” “children’s voices,”
and “announcements” in MAESTRO. “Dishes” in DESED corresponds to “cutlery and
dishes” in MAESTRO, and “dog” in DESED is a super-class for “dog bark”. This
mapping ensures the network to behave similarly for loss computation during
training in case of these mapped classes irrespective of their original
sources. However, this class mapping may cause a domain mismatch because the
audio features of these classes might differ significantly between the two
datasets.

To analyze these differences, we use a 2D convolutional neural network (CNN)
as convolutional layers applied to extract local invariant features. We denote
the input time-frequency representations of 2D CNNs as $X\in R^{F\times T}$,
where $F$ and $T$ are the numbers of frequency bins and frames, respectively.
With a batch size of $N$, we can represent feature maps as $M\in R^{N\times
C\times F\times T}$, where $C$ is the number of channels. Following [13], we
utilize instance statistics across a specific dimension, i.e., mean and
standard deviation (std), to analyze audio characteristics in 2D CNNs.

Specifically, the frequency-wise statistics can be formulated as:

$s^{(F)}=\textrm{Concat}(\mu(F),\sigma(F))$ (1)

where $\mu(F)\in R^{F}$ and $\sigma(F)\in R^{F}$ are mean and std computed
across $F$-axis, respectively. ‘Concat’ stands for concatenating the two
vectors.Similarly, the channel-wise statistics is obtained as:

$s^{(C)}=\textrm{Concat}(\mu(C),\sigma(C))$ (2)

where $\mu(C)\in R^{F}$ and $\sigma(C)\in R^{C}$ are mean and std calculated
across $C$-axis, respectively.

We compare $s^{(F)}$ and $s^{(C)}$ using 2D t-SNE visualization [16] in Figure
1, where the feature maps are generated by the CRNN baseline model of DCASE
2024 Task 4. It is observed that the features from different domains are
better separated with frequency-wise statistics than that with channel-wise
statistics, demonstrating that the frequency feature dimension carries more
domain-relevant information in comparison to the channel dimension.

(a) frequency-wise statistics

(b) channel-wise statistics

Figure 1: 2D t-SNE visualizations on the DCASE 2024 Task 4 dataset using the
feature map of CRNN.

###  2.2 Frequency-wise MixStyle

MixStyle [17] is a common DG method motivated by the observation that the
visual domain is closely related to image style. Specifically, MixStyle mixes
the feature statistics of two instances with a random convex weight to
simulate new styles. This helps the model generalize better across different
domains by increasing the diversity of the training data.

We analyzed the relationship between the domain and the statistics of each
feature dimension, which revealed that the frequency feature dimension carries
more domain-relevant information than the channel dimension. Therefore, we
propose to use Freq-MixStyle, which normalizes the frequency bands of
spectrograms and then denormalizes them with mixed frequency components from
two different recordings. The mixing coefficient specifies the shape of the
Beta distribution. More specifically, given an input batch $x$, Freq-MixStyle
first generates a reference batch $\hat{x}$ from $x$. When both datasets are
given in one single batch, $x$ is sampled from two different domains $d$
(DESED) and $m$ (MAESTRO), e.g., $x=[x_{d},x_{m}]$. Then, $\hat{x}$ is
obtained by swapping the position of $x_{d}$ and $x_{m}$ as shown in Figure 2,
followed by a shuffling operation along the batch dimension applied to each
batch. After shuffling, Freq-MixStyle computes the feature statistics,
$(\mu(F,x),\sigma(F,x))$ and $(\mu(F,\hat{x}),\sigma(F,\hat{x}))$. Here,
$\mu(F,x)$ and $\sigma(F,x)$ are the mean and standard deviation of each
instance from the $x$ across $F$-axis. Freq-MixStyle then generates a mixture
of feature statistics by:

$\alpha_{mix}=\lambda\mu(F,x)+(1-\lambda)\mu(F,\hat{x}),$ (3)
$\beta_{mix}=\lambda\sigma(F,x)+(1-\lambda)\sigma(F,\hat{x}),$ (4)

where $\lambda$ is an instance-specific, random weight sampled from the Beta
distribution. Finally, the mixture of feature statistics is applied to the
style-normalized $x$ as:

$\text{Freq-
MixStyle}(x,\hat{x})=\alpha_{mix}\odot\frac{x-\mu(F,x)}{\sigma(F,x)}+\beta_{mix}$
(5)

Freq-MixStyle for domain generalization is beneficial as it helps the SED
model learn more robust frequency-wise features by simulating new domains
during the training process. This increases the diversity of the training
data, allowing the SED model to perform effectively on unseen domains. The
effect of Freq-MixStyle is regularized by a second parameter $p$ that controls
the probability of Freq-MixStyle applied to a batch of recordings. We will
experiment with different values of this probability to explore its effects.

###  2.3 Adaptive residual normalization

Figure 2: Illustration of our proposed method. The orange and blue labels
denote DESED and MAESTRO dataset, respectively.

While Freq-MixStyle mixes the feature statistics of two instances to create
new styles and enhance domain generalization, we also employ adaptive residual
normalization (AdaResNorm) to focus on adjusting the normalization process.
The AdaResNorm is based on the frequency-instance normalization (FreqIN) which
generalizes the features on multiple domains by applying instance
normalization in the frequency dimension.

$\text{FreqIN(x)}=\frac{x-\mu(F,x)}{\sqrt{\sigma^{2}(F,x)+\epsilon}}$ (6)

where $\epsilon$ is an extremely small constant for numerical stability.
Moreover, residual normalization adds an identity path to FreqIN with a hyper-
parameter for compensating the information loss. Inspired by residual
normalization, we introduce adaptive residual normalization, as shown in:

$\text{AdaResNorm}(x)=(a\cdot x+(1-a)\cdot\text{FreqIN}(x))\cdot b+c$ (7)

Here $a$, $b$, and $c$ are trainable parameters for balancing, scaling, and
shifting, respectively. By adding trainable parameters to control the trade-
off between identity and FreqIN, the normalization behavior can be adaptively
adjusted based on the characteristics of the data and the requirements of the
domain. This can help mitigate the information loss that might occur during
the MixStyle process. Additionally, by adjusting the normalization dynamically
based on the data characteristics, AdaResNorm can enhance the robustness of
the model when dealing with the diverse and mixed domains created by Freq-
MixStyle. Thus, integrating AdaResNorm with Freq-MixStyle provides a more
effective domain generalization by leveraging the strengths of both methods.
It is noted that the adaptive residual normalization is inserted after the
first convolution layer.

###  2.4 Sound event bounding box-based post-processing

Inspired by bounding box predictions in image object detection [18], the SEBBs
[19] are one-dimensional bounding boxes defined by event onset time, event
offset time, sound class, and confidence. They represent sound event
candidates with a confidence score. The final SED is derived by applying
class-wise event-level thresholding to SEBBs’ confidences. For high
sensitivity or recall (few missed hits), a low detection threshold is used to
detect events even when the confidence of the system is low. For high
precision (few false alarms), a higher threshold detects only events with high
confidence. SEBBs allow controlling system sensitivity without affecting the
detection of an event’s onset and offset times, unlike frame-level
thresholding.

To address the mismatch between synthetic validation and real-world test
datasets, we split the 3,470 clips of the strongly annotated AudioSet to get
an additional real validation dataset (373 clips). We then adopt a change-
detection-based approach for SEBBs (cSEBBs) tuned on this real validation
dataset. This method calculates “delta” scores by filtering the signal and
identifies peaks and troughs as tentative onsets and offsets. We merge gaps
caused by minor signal variations by comparing scores with a predefined
threshold. After tuning filter length and thresholds on the real validation
set, cSEBBs are used as the post-processing in our system.

###  2.5 Training model with DG

The backbone model, inherited from previous DCASE Task 4 challenge editions
[5, 14], consists of a CRNN [3] that uses self-supervised features from the
pre-trained BEATs [20] model. We integrate the MAESTRO data and DESED data
(strong, weak, and unlabeled) in one batch. First, we shuffle and mix the
MAESTRO and DESED strong data using Freq-MixStyle. After the first 2D CNN
layer, we employ the AdaResNorm. The CRNN includes a 2D CNN encoder with 7
convolutional layers, batch normalization, gated linear units, and dropout,
followed by a bi-directional GRU (biGRU) layer. BEATs features are
concatenated with CNN features before the BiGRU layer. Average pooling adjusts
the BEATs features to match the CNN sequence length. Attention pooling then
derives clip-wise and frame-wise posteriors. During training, the BEATs model
remains frozen, and the mean-teacher framework [21, 22] leverages unlabeled
and weakly labeled data. The attention pooling uses the softmax over classes,
with values for unlabeled classes masked to negative infinity before applying
softmax. After we output the frame level prediction, we use the cSEBBs to
further improve the performance on the DESED dataset.

Table 1: Performance in PSDS and mpAUC of different single-systems on the
DESED development set (Dev-PSDS), DESED public evaluation set (PubEval-PSDS),
and MAESTRO validation set (mpAUC) including frequency-dynamic convolution
(FDY), domain generalization (DG), and adaptive residual normalization
(AdaResNorm). The joint score is based on the sum of PubEval-PSDS (cSEBBs) and
mpAUC.

System | Dev-PSDS | PubEval-PSDS (raw) | PubEval-PSDS (cSEBBs) | mpAUC | Joint score | Parameters
---|---|---|---|---|---|---
Baseline | 0.493 | 0.549 | - | 0.721 | 1.270 | 1.8M
Baseline + FDY | 0.508 | 0.596 | 0.601 | 0.728 | 1.329 | 3.4M
Baseline + MixStyle | 0.516 | 0.573 | 0.583 | 0.724 | 1.307 | 1.8M
Baseline + FDY + MixStyle | 0.520 | 0.596 | 0.603 | 0.737 | 1.340 | 3.4M
Baseline + MixStyle + AdaResNorm | 0.520 | 0.574 | 0.588 | 0.726 | 1.314 | 1.8M
Baseline + FDY + MixStyle + AdaResNorm | 0.526 | 0.598 | 0.604 | 0.739 | 1.343 | 3.4M

##  3 Experiment Setting

###  3.1 Datasets

The DCASE 2024 Challenge Task 4 comprises of two datasets for training a
single model for SED.

####  3.1.1 DESED dataset

The DESED dataset [11] consists of 10-second audio clips recorded in domestic
environments or synthesized to mimic such environments. It features sound
events from 10 classes: alarm bell ringing, blender, cat, dishes, dog,
electric shaver/toothbrush, frying, running water, speech, and vacuum cleaner.
The synthetic part of the dataset is generated with Scaper [23], using
foreground events from Freesound datasets and backgrounds from YouTube videos
and the Freesound subset of the MUSAN [24] dataset. The real-world recordings
come from AudioSet [25] and include a weakly annotated set (1,578 clips), an
unlabeled set (14,412 clips), and a strongly annotated portion (3,470 clips).

####  3.1.2 MAESTRO Real dataset

The MAESTRO Real dataset [10] consists of a development set (6,426 clips) and
an evaluation set of long-form real-world recordings. This dataset contains
multiple temporally strong annotated events with soft labels from 17 classes.
However, only 11 classes are evaluated in this challenge because the other 6
do not have confidence levels over 0.5. These evaluated classes are:
birds_singing, car, people_talking, footsteps, children_voices, wind_blowing,
brakes_squeaking, large_vehicle, cutlery_and_dishes, metro_approaching, and
metro_leaving. This data was annotated using crowdsourcing, combining
temporally-weak labeling with a sliding window approach to determine events’
temporal localization. To obtain soft labels, annotations from multiple
annotators are aggregated via MACE [26]. The recordings are taken from the TUT
Acoustic Scenes 2016 dataset [27] and are between 3 to 5 minutes long.

###  3.2 Implementation details

The audio clips are resampled to 16 kHz mono using Librosa and then segmented
with a window size of 2048 samples and a hop length of 256 samples. Short-time
Fourier transform and Mel-filters are used to create log-mel spectrograms
spanning 0 to 8 kHz. The clips shorter than 10 seconds are padded with
silence. A batch size of 60 is used, with the distribution: 1/5 MAESTRO, 1/10
synth, 1/10 synth+strong, 1/5 weak, and 2/5 unlabeled. The training process
includes 50 warmup epochs, up to 300 epochs, and an epoch decay of 100.
Gradient clipping is set at 5.0, the EMA factor for the mean teacher is 0.999,
and validation occurs every 10 epochs. The Adam optimizer with a learning rate
of 0.001 is used. An exponential warmup is applied for the first 50 epochs,
with no early stopping. The probability $p$ of apply Freq-MixStyle we set 0.5
and coefficient to the beta distribution we choose 0.6 in all experiments.

###  3.3 Metrics

The DCASE 2024 Task 4 considers two metrics for evaluation. PSDS [28, 29] is
computed based on event onset and offset times, which is only available for
DESED data and thereby only on that fraction of the evaluation set. For
MAESTRO, segment-based labels (one second) are provided, and the segment-based
mean (macro-averaged) partial area under the ROC curve (mpAUC) is used as the
primary metric, with a maximum FP-rate of 0.1. mpAUC is computed with respect
to hard labels (threshold = 0.5) for the 11 classes listed. It is noted that
the DESED and MAESTRO clips are anonymized and shuffled in the evaluation set
to prevent manual domain identification.

##  4 Result and Analysis

We are first interested in evaluating the impact of the proposed DG method
used on DESED dataset in terms of PSDS metric that measures how well the
system detects the sound events. From Table 1, we can observe that the
baseline system has a PSDS of 0.549 on the public evaluation set. Introducing
frequency-dynamic convolution (FDY) improves the performance by enhancing
generalization. Then, Freq-MixStyle further increases the performance by
improving data diversity and robustness. By combining FDY and Freq-MixStyle,
we obtain a better result, especially with cSEBBs post-processing. Further, on
applying AdaResNorm to Freq-MixStyle helps with normalization to improve the
PSDS. The best PSDS of 0.604 on the public evaluation set is obtained when all
the above methods are used, benefits from each method.

Then we analyze the results in terms of mpAUC, which measures performance on
the MAESTRO dataset as shown in Table 1. Similar to the trend we observed in
case of the DESED dataset, each method shows their contribution to enhance the
baseline performance on MAESTRO dataset. On introducing FDY, Freq-MixStyle and
AdaResNorm together, we obtain the best mpAUC of 0.739 on MAESTRO Real
validation set. This confirms our proposed DG method works effectively on both
DESED and MAESTRO datasets.

Finally, we compute the joint score that combines PubEval-PSDS and mpAUC
scores for each system as DCASE 2024 Challenge Task 4 considers the joint
score for benchmarking the systems. From Table 1, we observe that the joint
score significantly improves from 1.270 to 1.343 on introducing FDY, Freq-
MixStyle, AdaResNorm and cSEBBs to the baseline system, showcasing their
contributions. While FDY provides a larger improvement compared to Freq-
MixStyle (especially on DESED subset), it comes at the cost of increase in
model parameters around twice that in the baseline. On the contrary, Freq-
MixStyle is useful for developing low-complexity systems as it does not affect
on the model parameters.

##  5 Conclusion

This work demonstrates the effectiveness of DG-based techniques for SED
systems trained with heterogeneous data. By integrating Freq-MixStyle,
adaptive residual normalization, and cSEBBs within a mean-teacher framework,
our approach successfully adapts mel-spectrograms and generalizes features
across multiple domains. Studies on the DCASE 2024 Challenge Task 4 dataset
shows that our DG-based method significantly improves both the polyphonic SED
score on the DESED dataset and the macro-average pAUC on the MAESTRO Real
dataset, outperforming the challenge baseline. These results highlight the
potential of DG-based methods in advancing SED adaptability to real-world
scenarios.

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# Quasi-Fuchsian vs Negative curvature metrics
on surface groups

Ethan Fricker  University of Illinois at Chicago<EMAIL_ADDRESS>and  Alex
Furman  University of Illinois at Chicago<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We compare two families of left-invariant metrics on a surface group
$\Gamma=\pi_{1}(\Sigma)$ in the context of course-geometry. One family comes
from Riemannian metrics of negative curvature on the the surface $\Sigma$, and
another from quasi-Fuchsian representations of $\Gamma$. We show that the
Teichmuller space $\operatorname{\mathscr{T}}(\Sigma)$ is the only common part
of these two families, even when viewed from the coarse-geometric perspective.

_To Benjy Weiss with gratitude and admiration_

##  1\. Introduction and Statement of the main result

###  1.A. Introduction and Background

Let $\Sigma$ be a closed surface of genus at least two, and
$\Gamma=\pi_{1}(\Sigma)$ its fundamental group. The Teichmuller space
$\operatorname{\mathscr{T}}(\Sigma)$ has several equivalent descriptions: as
the moduli space of (i) complex structures, or (ii) conformal structures, or
(iii) Riemannian structures of constant curvature $-1$ on $\Sigma$, or as (iv)
the space of discrete cocompact representations
$\Gamma\to\operatorname{PSL}_{2}({\mathbf{R}})$, up to conjugation. The latter
two points of view can be extended as follows:

  * •

$\mathscr{R}(\Sigma)$ – the space of all Riemannian structures of possibly
variable negative curvature, up to isotopy and scaling.

  * •

$\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$ – the space of all convex
cocompact representations

$\Gamma=\pi_{1}(\Sigma){\buildrel\over{\longrightarrow}}\operatorname{PSL}_{2}({\mathbf{C}})\cong\operatorname{Isom}^{+}({\mathbf{H}}^{3}),$

up to conjugation.

Both $\mathscr{R}(\Sigma)$ and $\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$
arise from convex cocompact isometric $\Gamma$-actions on CAT(-1) spaces: the
$\Gamma$-action by deck transformations on the universal cover
$(\tilde{\Sigma},d_{\tilde{g}})$ in the Riemannian case, and the
$\Gamma$-action on ${\mathbf{H}}^{3}$ in the quasi-Fuchsian case.

We can put these notions into an even broader context by looking at the space
$\operatorname{\mathscr{D}_{\Gamma}}$ of equivalence classes $[d]$ of left-
invariant metrics $d$ on $\Gamma$ obtained from restricting the metric of the
underlying Gromov-hyperbolic space to a $\Gamma$-orbit. Here two metrics
$d,d^{\prime}$ on $\Gamma$ are equivalent if they are bounded distance from
each other after scaling:

$d\sim d^{\prime}\qquad\textrm{if}\qquad\exists
k,A:\quad|d^{\prime}(\gamma_{1},\gamma_{2})-k\cdot
d(\gamma_{1},\gamma_{2})|\leq A.$

This perspective, introduced by the second author in [Fur] (see also more
recent treatment in Bader–Furman [BF]), allows to observe possible
”geometries” of $\Sigma$ from the ”outside” by studying the corresponding
classes $[d]\in\operatorname{\mathscr{D}_{\Gamma}}$ of metrics $d$ on
$\Gamma$. The space $\operatorname{\mathscr{D}_{\Gamma}}$ can be defined for a
general non-elementary Gromov hyperbolic group $\Gamma$, and
$\operatorname{\mathscr{D}_{\Gamma}}$ contains classes of metrics on $\Gamma$
from various sources, such as word metrics on $\Gamma$, Green metrics
associated with symmetric generating random walks on $\Gamma$ (see
Blachère–Haïssinsky–Mathieu [BHM1, BHM2]), Anosov representations of $\Gamma$
in higher rank simple Lie groups (see Dey–Kapovich [DK]), etc.

To avoid ambiguity in scaling we can normalize metrics $d$ by the growth

$h_{d}=\lim_{R\to\infty}\frac{1}{R}\log\\#\left\\{{\gamma\in\Gamma}\ |\
{d(\gamma,e)<R}\right\\},$

replacing $d$ by $\hat{d}=h_{d}\cdot d$, so that $h_{\hat{d}}=1$. For
$\delta\in\operatorname{\mathscr{D}_{\Gamma}}$ we can define:

  * •

Marked Length Spectrum $\ell_{\delta}:\Gamma\to{\mathbf{R}}_{+}$ given by the
limit

$\ell_{\delta}\left(\gamma\right)=\lim_{n\to\infty}\frac{\hat{d}(\gamma^{n},e)}{n}$

where $\delta=[d]$ and $\hat{d}=h_{d}\cdot d$. Note that $\ell_{\delta}$ is
constant on conjugacy classes, so we can write it as
$\ell_{\delta}:\mathcal{C}_{\Gamma}\to{\mathbf{R}}_{+}$.

  * •

Patterson-Sullivan-like $\Gamma$-invariant measure class $[\nu_{\delta}^{\rm
PS}]$ on $\partial\Gamma$ (see Coorneart [Coor], and [Fur, BF]).

  * •

Bowen–Margulis–Sullivan-like $\Gamma$-invariant Radon measure $m_{\delta}^{\rm
BMS}$ on the space $\partial^{(2)}\Gamma$ of distinct pairs $(\xi,\eta)$ of
points on $\partial\Gamma$ (see [Fur, BF]).

In [Fur] (see also Bader–Furman [BF]), it was shown that each
$\delta\in\operatorname{\mathscr{D}_{\Gamma}}$ is determined by each of these
objects. Furthermore, extending a prior work of Bader–Muchnik [BM], Garncarek
[Garn] showed that for each $\delta\in\operatorname{\mathscr{D}_{\Gamma}}$ the
quasi-regular unitary $\Gamma$-representation

$\pi_{\delta}:\Gamma{\buildrel\over{\longrightarrow}}U(\partial\Gamma,\nu_{\delta}^{\rm
PS})$

is irreducible, and that the map
$\operatorname{\mathscr{D}_{\Gamma}}{\buildrel\over{\longrightarrow}}\hat{\Gamma}$,
$\delta\mapsto\pi_{\delta}$, is also injective. Thus
$\operatorname{\mathscr{D}_{\Gamma}}$ can be embedded into any obe of the the
following spaces:

${\mathbf{R}}_{+}^{\mathcal{C}_{\Gamma}},\qquad\operatorname{Prob}(\partial\Gamma),\qquad{\rm
Meas}_{\Gamma}\left(\partial^{(2)}\Gamma\right),\qquad\hat{\Gamma}.$

The space $\operatorname{\mathscr{D}_{\Gamma}}$ is also equipped with a
natural metric: given two classes $\delta=[d]$, $\delta^{\prime}=[d^{\prime}]$
in $\operatorname{\mathscr{D}_{\Gamma}}$ we can define the (log) Lipschitz
distance by

$\rho_{\operatorname{Lip}}\left({\delta},{\delta^{\prime}}\right):=\log\left(\inf\left\\{{\frac{K}{k}}\
|\ {\exists A,\ k\cdot d-A\leq d^{\prime}\leq K\cdot d+A}\right\\}\right).$

It is clear from the definition that
$\rho_{\operatorname{Lip}}\left({-},{-}\right)$ is symmetric and satisfies the
triangle inequality. One can see that for any $a,b\in\Gamma\setminus\\{e\\}$
one has

$\left|\log\left(\frac{\ell_{\delta}(a)}{\ell_{\delta}(b)}:\frac{\ell_{\delta^{\prime}}(a)}{\ell_{\delta^{\prime}}(b)}\right)\right|\leq\rho_{\operatorname{Lip}}\left({\delta},{\delta^{\prime}}\right).$

This shows that
$\rho_{\operatorname{Lip}}\left({\delta},{\delta^{\prime}}\right)=0$ implies
$\ell_{\delta}=\ell_{\delta^{\prime}}$, which occurs only when
$\delta=\delta^{\prime}$. So $\rho_{\operatorname{Lip}}\left({},{}\right)$ is
indeed a metric on $\operatorname{\mathscr{D}_{\Gamma}}$ (see also a recent
work of Cantrell–Tanaka [Cantrell+Tanaka] for a more detailed picture).

###  1.B. Riemannian and Quasi-Fuchsian structures on Surfaces

In this paper we focus on surface group $\Gamma=\pi_{1}(\Sigma)$ and two
specific sources for $\delta\in\operatorname{\mathscr{D}_{\Gamma}}$: namely
$\mathscr{R}(\Sigma)$ and $\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$.

For the case of negatively curved Riemannian metric $g$ on $\Sigma$, fix
$x\in\tilde{\Sigma}$ and consider the metric on $\Gamma$

$d_{g,x}(\gamma_{1},\gamma_{2}):=d_{\tilde{g}}(\gamma_{1}x,\gamma_{2}x).$

Since $|d_{g,x}-d_{g,x^{\prime}}|\leq\operatorname{diam}(\Sigma,g)$ the class
$[d_{g,x}]$ does not depend on the choice of $x\in\tilde{\Sigma}$, and we can
denote this class by $\delta_{g}=[d_{g,x}]$. Note that $h_{d_{g,x}}$ is the
topological entropy of the geodesic flow on the unit tangent bundle
$T^{1}\Sigma$ to $\Sigma$, and we assume that all $g\in\mathscr{R}(\Sigma)$
are normalized so that $h_{d_{g,x}}=1$. We have a map

(1.1) $i:\mathscr{R}(\Sigma)\ {\buildrel\over{\longrightarrow}}\
\operatorname{\mathscr{D}_{\Gamma}}.$

The Marked Length Spectrum Rigidity Conjecture, that for surfaces was proved
by Otal [Otal] and Croke [Croke], asserts that a Riemannian structure $g$ of
variable negative curvature on a surface $\Sigma$ is uniquely determined by
the function $\ell_{g}:\mathcal{C}_{\Gamma}\to{\mathbf{R}}$. As a consequence,
we obtain:

######  Proposition 1.1.

The map
$\mathscr{R}(\Sigma){\buildrel\over{\longrightarrow}}\operatorname{\mathscr{D}_{\Gamma}}$,
$i:g\mapsto\delta_{g}$, is injective.

Our second source of examples, are quasi-Fuchsian representations. For
$q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$ choose a representation
$\pi:\Gamma\to\operatorname{Isom}^{+}({\mathbf{H}}^{3})\cong\operatorname{PSL}_{2}({\mathbf{C}})$
in this class and a point $y\in{\mathbf{H}}^{3}$ and consider the metric on
$\Gamma$:

$d_{\pi,y}(\gamma_{1},\gamma_{2}):=d_{{\mathbf{H}}^{3}}\left(\pi(\gamma_{1}).y,\pi(\gamma_{2}).y\right).$

The class $[d_{\pi,y}]$ does not depend on the choice of
$y\in{\mathbf{H}}^{3}$ and remains unchanged if $\pi$ is replaced by a
conjugate $\gamma\mapsto g\pi(\gamma)g^{-1}$; thus we write $\delta_{q}$ for
$[d_{\pi,y}]$. This gives a well defined map

(1.2) $j:\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)\
{\buildrel\over{\longrightarrow}}\ \operatorname{\mathscr{D}_{\Gamma}}.$

One can deduce from a work of Burger [Burger] (or Dal’bo–Kim [Dalbo+Kim]), the
following.

######  Proposition 1.2.

The map
$\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma){\buildrel\over{\longrightarrow}}\operatorname{\mathscr{D}_{\Gamma}}$,
$j:q\mapsto\delta_{q}$, is injective.

Hence one might view each of $\mathscr{R}(\Sigma)$ and
$\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$ as being embedded in
$\operatorname{\mathscr{D}_{\Gamma}}$.

######  Remark 1.3.

We note in passing that the uniformization theorem allows us to view
$\mathscr{R}(\Sigma)$ as a bundle over $\operatorname{\mathscr{T}}(\Sigma)$
with fibers that can be identified with the positive cone
$C^{\infty}_{+}(\Sigma)/{\mathbf{R}}_{+}$; in particular $\mathscr{R}(\Sigma)$
is connected. One can show that the map (1.1) is continuous, and so the image
$i(\mathscr{R}(\Sigma))$ in $\operatorname{\mathscr{D}_{\Gamma}}$ is
connected.

Ahlfors and Bers showed that $\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$
can be identified with
$\operatorname{\mathscr{T}}(\Sigma)\times\operatorname{\mathscr{T}}(\Sigma)$,
and is in particular connected. The map (1.2) can be shown to be continuous;
hence the image $j(\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma))$ is a
connected subset of $\operatorname{\mathscr{D}_{\Gamma}}$.

It is natural to wonder whether the intersection

$i(\mathscr{R}(\Sigma))\cap
j(\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma))\subset\operatorname{\mathscr{D}_{\Gamma}}$

contains anything except for the image of
$\operatorname{\mathscr{T}}(\Sigma)$. In other words, is it true that given a
quasi-Fuchsian representation
$\pi:\Gamma{\buildrel\over{\longrightarrow}}\operatorname{PSL}_{2}({\mathbf{C}})$
and a negatively curved metric $g$ on the surface $\Sigma$, there exist
constants $k,A$ and points $x\in\tilde{\Sigma}$, $y\in{\mathbf{H}}^{3}$, so
that

$k\cdot d_{\tilde{g}}(\gamma.x,x)-A\leq
d_{{\mathbf{H}}^{3}}(\pi(\gamma).y,y)\leq k\cdot
d_{\tilde{g}}(\gamma.x,x)+A\qquad(\gamma\in\Gamma)$

only if $g$ has constant curvature, $\pi$ is conjugate into
$\operatorname{PSL}_{2}({\mathbf{R}})$, and $(\Sigma,g)$ and $\pi$ represent
the same point in $\operatorname{\mathscr{T}}(\Sigma)$?

Our main result answers this affirmatively.

######  Theorem A.

The images of $\mathscr{R}(\Sigma)$ and
$\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$ in
$\operatorname{\mathscr{D}_{\Gamma}}$ have only
$\operatorname{\mathscr{T}}(\Sigma)$ in common. Moreover, for any
$q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)\setminus\operatorname{\mathscr{T}}(\Sigma)$
there is $\alpha_{q}>0$ so that

$\rho_{\operatorname{Lip}}\left({\delta_{q}},{\delta_{g}}\right)\geq\alpha_{q}>0$

for all $g\in\mathscr{R}(\Sigma)$.

The following natural question remains open.

######  Question 1.4.

Is it true that for
$g\in\mathscr{R}(\Sigma)\setminus\operatorname{\mathscr{T}}(\Sigma)$ there
exists $\beta_{g}>0$ so that

$\rho_{\operatorname{Lip}}\left({\delta_{q}},{\delta_{g}}\right)\geq\beta_{g}>0$

for all $q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$?

### Acknowledgement

This note is dedicated to Benjy Weiss on the occasion of his $80$th birthday.
His profound contributions to Ergodic Theory and Dynamics, breadth of his
interests and originality of his ideas are an inspiration to us and many, many
others.

We would like to thank Dick Canary for the useful discussion about quasi-
fuchsian representations.

A.F. acknowledges the support of the NSF grant DMS-2005493 and BSF
grant-2018258.

##  2\. Length inequalities for negatively curved surfaces

Consider the topological picture first. Let $\Sigma$ be a closed surface of
genus at least two, $\Gamma=\pi_{1}(\Sigma)$ the corresponding surface group,
that acts on the universal cover $\tilde{\Sigma}$ by deck transformations.
This action extends to the action of $\Gamma$ on the boundary circle
$\partial\tilde{\Sigma}$, which is also the Gromov boundary $\partial\Gamma$
of $\Gamma$. Every $\gamma\neq 1$ in $\Gamma$ has two fixed points on the
topological circle $\partial\tilde{\Sigma}$: a repelling point $\gamma^{-}$
and an attracting point $\gamma^{+}$. We shall consider a pair $a,b\in\Gamma$
where $a^{-}$, $a^{+}$, $b^{-}$, $b^{+}$ are four distinct points on the
circle.

Let $A=(\alpha_{1},\alpha_{2})$ and $B=(\beta_{1},\beta_{2})$ be two ordered
pairs on a circle $C$, where all four points are distinct. The action of
$\operatorname{Homeo}(C)$ on such pairs has $3$ orbits corresponding to $3$
possible relative positions of the two pairs $A$, $B$:

  * •

The pairs are linked, meaning that $\beta_{1}$ and $\beta_{2}$ lie in distinct
arcs defined by $\\{\alpha_{1},\alpha_{2}\\}$ – connected components of
$C\setminus\\{\alpha_{1},\alpha_{2}\\}$. The relation of being linked is
symmetric: $A$ is linked with $B$ iff $B$ is linked with $A$. The order within
the pairs $A=(\alpha_{1},\alpha_{2})$ and $B=(\beta_{1},\beta_{2})$ does not
change the status of being linked. We say that disjoint pairs $A$ and $B$ are
unlinked if they are not linked.

  * •

The pairs $A$ and $B$ are unlinked and aligned, if in the arc
$\widearc{\alpha_{1},\alpha_{2}}$ determined by $\\{\alpha_{1},\alpha_{2}\\}$
on $C$ containing $\beta_{1}$ and $\beta_{2}$ one has linear order
$\alpha_{1}<\beta_{1}<\beta_{2}<\alpha_{2}$. We note that $A$ is unlinked and
aligned with $B$ iff $B$ is unlinked and aligned with $A$. In this case
flipping the order in both pairs $A$ and $B$ simultaneously does not change
the status of being aligned.

  * •

The pairs $A$ and $B$ are unlinked and misaligned, if in the arc
$\widearc{\alpha_{1},\alpha_{2}}$ determined by $\\{\alpha_{1},\alpha_{2}\\}$
on $C$ containing $\beta_{1}$ and $\beta_{2}$ one has linear order
$\alpha_{1}<\beta_{2}<\beta_{1}<\alpha_{2}$. We note that $A$ is unlinked and
misaligned with $B$ iff $B$ is unlinked and misaligned with $A$. In this case
flipping the order in both of $A$ and $B$ simultaneously does not change the
status of being misaligned. Yet flipping the order in either $A$ or $B$ makes
the pair unlinked and aligned.

$\alpha_{1}$$\alpha_{2}$$\beta_{1}$$\beta_{2}$ linked

$\beta_{1}$$\beta_{2}$$\alpha_{1}$$\alpha_{2}$ unlinked and aligned

$\beta_{2}$$\beta_{1}$$\alpha_{1}$$\alpha_{2}$ unlinked and misaligned

Let us now choose a negatively curved Riemannian metric $g$ on $\Sigma$, and
let $\tilde{g}$ be its lift to $\tilde{\Sigma}$. Denote by $d_{\tilde{g}}$ the
corresponding distance on $\tilde{\Sigma}$, and by
$\ell_{g}:\Gamma\to[0,\infty)$ the associated stable length

$\ell_{g}(\gamma):=\lim\frac{1}{n}d_{\tilde{g}}(\gamma^{n}.p,p)$

where $p\in\tilde{\Sigma}$ is arbitrary.

######  Theorem 2.1.

Let $a,b\in\Gamma$ be non-trivial elements with distinct fixed points
$a-_{,}a^{+},b^{-},b^{+}$ on the boundary circle $\partial\Gamma$. Then

  1. (1)

If $(a^{-},a^{+})$ and $(b^{-},b^{+})$ are linked, then

$\ell_{g}(ab)<\ell_{g}(a)+\ell_{g}(b).$

  2. (2)

If $(a^{-},a^{+})$ and $(b^{-},b^{+})$ are unlinked and aligned, then

$\ell_{g}(ab)>\ell_{g}(a)+\ell_{g}(b).$

  3. (3)

If $(a^{-},a^{+})$ and $(b^{-},b^{+})$ are unlinked and misaligned, then

$\ell_{g}(a^{-1}b)>\ell_{g}(a)+\ell_{g}(b).$

###### Proof.

First recall that in the case of negatively curved manifolds, such as
$(\Sigma,g)$ the stable length $\ell_{g}(\gamma)$ can also be defined as the
_minimal translation length_

$\ell_{g}(\gamma)=\inf_{p\in\tilde{\Sigma}}d_{\tilde{g}}(\gamma.p,p).$

Moreover, when $\ell_{g}(\gamma)>0$, which is the case of any non-trivial
$\gamma\neq 1$, the $\inf$ is attained and the set

$\operatorname{Ax}_{{\gamma}}:=\left\\{{p\in\tilde{\Sigma}}\ |\
{d_{\tilde{g}}(\gamma.p,p)=\ell_{g}(\gamma)}\right\\}$

is the geodesic line $(\gamma^{-},\gamma^{+})$ in $\tilde{\Sigma}$. It is
called the axis of $\gamma$.

Elementary topology of the disc $\tilde{\Sigma}$ implies that when
$(a^{-},a^{+})$ and $(b^{-},b^{+})$ are linked, the axes
$\operatorname{Ax}_{{a}}$ and $\operatorname{Ax}_{{b}}$ must intersect in
$\tilde{\Sigma}$. Due to negative curvature the intersection is a singleton:
$\operatorname{Ax}_{{a}}\cap\operatorname{Ax}_{{b}}=\\{p\\}$. Since
$p\in\operatorname{Ax}_{{b}}$, we have $x=b^{-1}.p\in\operatorname{Ax}_{{b}}$.
Similarly, we have $p$ and $y=a.p$ are in $\operatorname{Ax}_{{a}}$ as well.
To prove part $(1)$ we use the triangle inequality to obtain for $x=b^{-1}.p$:

$\begin{split}\ell_{g}(ab)\leq
d_{\tilde{g}}(x,ab.x)&<d_{\tilde{g}}(x,b.x)+d_{\tilde{g}}(b.x,ab.x)\\\
&=d_{\tilde{g}}(b^{-1}.p,p)+d_{\tilde{g}}(p,a.p)=\ell_{g}(b)+\ell_{g}(a)\end{split}$

We observe that the second inequality is strict and will sharpen it in the
proof of Theorem A below.

$a^{-}$$a^{+}$$b^{-}$$b^{+}$$p$$x=b^{-1}.p$ $y=ab.x$ linked

$b^{-}$$b^{+}$$a^{-}$$a^{+}$$(ab)^{-}$$(ab)^{+}$$b.(ab)^{-}$$b.(ab)^{+}$$p$$x=b^{-1}.p$$ab.x$
unlinked and aligned

In the case where the pairs $(a^{-},a^{+})$ and $(b^{-},b^{+})$ are unlinked
and aligned, we remind ourselves of the definition, that $a^{-},a^{+}$ define
an arc $\widearc{a^{-}a^{+}}$ on the boundary circle containing both $b^{-}$
and $b^{+}$, which can be equipped with a linear order (anti-clockwise in the
figure) so that

$a^{-}<b^{-}<b^{+}<a^{+}.$

The action of $b$ on the arc/interval from $b^{+}$ to $a^{+}$ is decreasing
towards the fixed point $b^{+}$, while the action of $a$ is increasing towards
$a^{+}$. Thus $ab$ maps this interval into itself, and therefore the
attracting point $(ab)^{+}$ satisfies $b^{+}<(ab)^{+}<a^{+}$. Moreover, we
have

$b^{+}=b.b^{+}<b.(ab)^{+}<(ab)^{+}.$

Since the repelling fixed point of an element is the attracting fixed point of
it’s inverse, the same argument gives $a^{-}<(ab)^{-}<b^{-}$. We claim that
$a^{-}<b.(ab)^{-}<(ab)^{-}$. Indeed, in the linear order on the arc
$\widearc{b^{+}b^{-}}$ that contains $a^{\pm}$ so that
$b^{+}<a^{+},a^{-}<b^{-}$ the map $b$ is decreasing, and thus
$\xi=b.(ab)^{-}<(ab)^{-}$. Since $a.\xi=(ab).(ab)^{-}=(ab)^{-}>\xi$ we deduce
that $a^{-}<\xi<(ab)^{-}$. Hence

$a^{-}<b.(ab)^{-}<(ab)^{-}.$

We conclude that pair $\left((ab)^{-},(ab)^{+}\right)$ is linked with its
image under $b$. Denote by $p$ the intersection of $\operatorname{Ax}_{{ab}}$
and $b.\operatorname{Ax}_{{ab}}$ in $\tilde{\Sigma}$, and let $x=b^{-1}.p$.
Since $p\in b.\operatorname{Ax}_{{ab}}$ we have $x\in\operatorname{Ax}_{{ab}}$
and $ab.x\in\operatorname{Ax}_{{ab}}$ as well. Thus the points $x$, $p=b.x$,
$ab.x=a.p$ lie on the geodesic line $\operatorname{Ax}_{{ab}}$, and in fact in
this linear order. This can be seen by inspecting the projections of these
points to $\operatorname{Ax}_{{a}}$ and $\operatorname{Ax}_{{b}}$, making use
of the assumption that the pairs are aligned. Hence

$\ell_{g}(ab)=d_{\tilde{g}}(x,ab.x)=d_{\tilde{g}}(x,b.x)+d_{\tilde{g}}(p,a.p)>\ell_{g}(b)+\ell_{g}(a).$

The strict inequality here occurs because $p\not\in\operatorname{Ax}_{{a}}$
and $x\not\in\operatorname{Ax}_{{b}}$. This proves statement (2).

Statement (3) follows from (2) by replacing $a$ by $a^{-1}$. This completes
the proof of Theorem 2.1. ∎

##  3\. Spiraling of the boundary of a quasi-fuchsian embedding

Let $\Gamma=\pi_{1}(\Sigma)$ be a surface group, and
$q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)$ be defined by a
representation $\pi:\Gamma\to\operatorname{PSL}_{2}({\mathbf{C}})$. For
$\gamma\in\Gamma$ the element
$g=\pi(\gamma)\in\operatorname{PSL}_{2}({\mathbf{C}})$ has two preimages
$\pm\hat{g}$ in $\operatorname{SL}_{2}({\mathbf{C}})$. Since the traces
$\pm\operatorname{tr}(\hat{g})$ are invariant under conjugation, we can denote
them by $\pm\operatorname{tr}_{q}(\gamma)$. The following is a particular case
of a lemma of Vinberg [Vinberg] (see [MR]*Corollary 3.2.5)

######  Lemma 3.1.

Let $\Gamma=\pi_{1}(\Sigma)$ a surface group, and
$q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)\setminus\operatorname{\mathscr{T}}(\Sigma)$.
Then there exists $\gamma\in\Gamma$ with $\
\pm\operatorname{tr}_{q}(\gamma)\in{\mathbf{C}}\setminus{\mathbf{R}}$.

Let $\pi:\Gamma\to\operatorname{PSL}_{2}({\mathbf{C}})$ be a quasi-Fuchsian
representation. There exists a $\Gamma$-equivariant continuous map

$\phi:\partial\Gamma{\buildrel\over{\longrightarrow}}\mathbb{P}_{\mathbf{C}}^{1},\qquad\phi\circ\gamma=\pi(\gamma)\circ\phi$

that is a homeomorphism between the topological circle $\partial\Gamma$ and
the Jordan curve on the sphere $\mathbb{P}_{\mathbf{C}}^{1}$ formed by the
limit set $L_{\pi(\Gamma)}$ of $\pi(\Gamma)$.

######  Proposition 3.2.

Let
$q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)\setminus\operatorname{\mathscr{T}}(\Sigma)$
be given by a quasi-Fuchsian representation
$\pi:\Gamma{\buildrel\over{\longrightarrow}}\operatorname{PSL}_{2}({\mathbf{C}})$.
Then there exists an isometrically embedded hyperbolic plane
${\mathbf{H}}^{2}\subset{\mathbf{H}}^{3}$ and a sequence
$\xi_{1},\xi_{2},\dots\to\xi_{*}\in\partial\Gamma$ whose cyclic order with
respect to the circle $\partial\Gamma$ is

$\xi_{1},\ \xi_{2},\ \xi_{3},\ \xi_{4},\ \dots,\ \xi_{*}$

and whose images $\phi(\xi_{n})\in\mathbb{P}_{\mathbf{C}}^{1}$ lie on the
boundary circle $\partial{\mathbf{H}}^{2}$ in the following cyclic order:

$\phi(\xi_{1}),\ \phi(\xi_{3}),\ \phi(\xi_{5}),\ \dots,\ \phi(\xi_{*}),\
\dots\ \phi(\xi_{6}),\ \phi(\xi_{4}),\ \phi(\xi_{2}).$

In particular, we have

  * •

$(\xi_{1},\xi_{4})$ and $(\xi_{2},\xi_{3})$ are unlinked and aligned in
$\partial\Gamma$, while $(\phi(\xi_{1}),\phi(\xi_{4}))$ and
$(\phi(\xi_{2}),\phi(\xi_{3}))$ are linked in $\partial{\mathbf{H}}^{2}$.

  * •

$(\xi_{1},\xi_{3})$ and $(\xi_{2},\xi_{4})$ are linked in $\partial\Gamma$,
while $(\phi(\xi_{1}),\phi(\xi_{3}))$ and $(\phi(\xi_{2}),\phi(\xi_{4}))$ are
unlinked and aligned in $\partial{\mathbf{H}}^{2}$.

###### Proof.

Fix an element $\gamma\in\Gamma$ with
$\pm\operatorname{tr}_{q}(\gamma)\in{\mathbf{C}}\setminus{\mathbf{R}}$ as in
Lemma 3.1. Note that $\gamma$ must be hyperbolic, and denote by $\xi_{*}$ the
attracting point $\gamma^{+}\in\partial\Gamma$. At the same time
$\pi(\gamma)\in\operatorname{PSL}_{2}({\mathbf{C}})$ is loxodromic with an
attracting point $\phi(\gamma^{+})$. Identifying $\mathbb{P}_{\mathbf{C}}^{1}$
with ${\mathbf{C}}\cup\\{\infty\\}$ and replacing
$\pi:\Gamma{\buildrel\over{\longrightarrow}}\operatorname{PSL}_{2}({\mathbf{C}})$
by an appropriate conjugate we may assume $\phi(\gamma^{+})=\infty$ and
$\phi(\gamma^{-})=0$. Then the action of $\pi(\gamma)$ on ${\mathbf{C}}$ is
given by the linear map

$z\mapsto(\lambda e^{2\pi i\theta})\cdot
z\qquad\textrm{with}\qquad\lambda>1,\qquad\theta\in{\mathbf{R}}\setminus{\mathbf{Z}}.$

Identify $\partial\Gamma\setminus\\{\gamma^{+}\\}$ with ${\mathbf{R}}$ so that
$\gamma^{-}$ corresponds to $0\in{\mathbf{R}}$. With a slight abuse of
notation we write $\gamma$ and $\phi$ for the corresponding homeomorphism of
${\mathbf{R}}$, and an equivariant injective continuous map
${\mathbf{R}}{\buildrel\over{\longrightarrow}}{\mathbf{C}}$. Note that
$\gamma(0)=0$, $\gamma$ is strictly increasing on $[0,\infty)$ (and strictly
decreasing on $(-\infty,0]$), while $\phi$ satisfies

(3.1) $\phi(\gamma(t))=\lambda e^{2\pi i\theta}\cdot\phi(t)$

and $|\phi(t)|\to\infty$ as $|t|\to\infty$. Since $\phi(t)\neq 0$ for all
$t\in(0,\infty)$ there exist continuous functions $r:(0,\infty)\to(0,\infty)$
and $s:(0,\infty)\to{\mathbf{R}}$ so that

$\phi(t)=r(t)\cdot e^{2\pi i\cdot s(t)}\qquad(t>0).$

Thus (3.1) implies that

$r(\gamma^{n}(t))=\lambda^{n}\cdot r(t),\qquad s(\gamma^{n}(t))=s(t)+n\Theta$

where $\Theta\in\theta+{\mathbf{Z}}$. Note that the assumption that
$q\in\operatorname{\mathscr{Q}\mathscr{F}}(\Sigma)\setminus\operatorname{\mathscr{T}}(\Sigma)$
gives $\theta\not\in{\mathbf{Z}}$ (Lemma 3.1), implying $\Theta\neq 0$.

Fix $t_{0}>0$ and use points $t_{n}=\gamma^{n}(t_{0})$, $n\in{\mathbf{Z}}$, to
partition the ray $(0,\infty)$. Let

$R_{0}=\max\left\\{{r(t)}\ |\ {0\leq t\leq t_{0}}\right\\},\qquad
r_{0}=\min\left\\{{r(t)}\ |\ {t_{0}\leq t<\infty}\right\\}.$

Then $|\phi(t)|=r(t)\leq\lambda^{n}\cdot R_{0}$ for all $t\in[0,t_{n}]$, and
$|\phi(t)|=r(t)\geq\lambda^{n}\cdot r_{0}$ for all $t\geq t_{n}$. We can now
choose integers $n(1)<m(1)<n(2)<m(2)<\dots$ so that

$(m(k)-n(k))\cdot|\Theta|>1,\qquad\lambda^{n(k+1)-m(k)}>R_{0}/r_{0}$

for all $k\in{\mathbf{N}}$. The first condition guarantees that
$s(t_{m(k)})>s(t_{n(k)})+1$ and therefore there exist

$\xi_{k}\in[t_{n(k)},t_{m(k)}]\qquad\textrm{with}\qquad e^{2\pi i\cdot
s(\xi_{k})}=(-1)^{k}.$

Thus $\phi(\xi_{k})=(-1)^{k}r(\xi_{k})$ lie on the real line
${\mathbf{R}}\subset{\mathbf{C}}$ on both sides of $0\in{\mathbf{R}}$ in
alternating order. Since $\xi_{k}\leq t_{m(k)}<t_{n(k+1)}\leq\xi_{k+1}$ we
also have

$|\phi(\xi_{k})|=r(\xi_{k})\leq\lambda^{m(k)}\cdot R_{0}<\lambda^{n(k+1)}\cdot
r_{0}\leq r(\xi_{k+1})=|\phi(\xi_{k+1})|.$

Thus the sequence $\\{|\phi(\xi_{k})|\\}$ is monotonically increasing. In
particular, we have

$\dots\phi(\xi_{5})<\phi(\xi_{3})<\phi(\xi_{1})<0<\phi(\xi_{2})<\phi(\xi_{4})<\phi(\xi_{6})<\dots$

on ${\mathbf{R}}\subset{\mathbf{C}}$. Recalling that $\phi(\xi_{*})=\infty$ we
get the required cyclic order. ∎

##  4\. Proof of Theorem A

Let us first recall two general well known facts, one related to CAT(-1)
spaces $(X,d_{X})$, and another to Gromov hyperbolic groups $\Gamma$ acting on
their boundary $\partial\Gamma$. We will apply them to $X={\mathbf{H}}^{3}$
and to the surface group $\Gamma=\pi_{1}(\Sigma)$.

Recall that given a point $p\in{\mathbf{H}}^{3}$ and a pair of distinct
boundary points $\xi\neq\eta\in\partial{\mathbf{H}}^{3}$ the Busemann function
is given by the limit

$\operatorname{B}_{p}(\xi,\eta)=\lim_{x\to\xi,\
y\to\eta}\left(d_{{\mathbf{H}}^{3}}(p,x)+d_{{\mathbf{H}}^{3}}(p,y)-d_{{\mathbf{H}}^{3}}(x,y)\right).$

Triangle inequality implies that $\operatorname{B}_{p}(\xi,\eta)\geq 0$.
Crucial for our purposes, is the fact that the strict inequality occurs unless
$p$ lies on the geodesic line $(\xi,\eta)$:

$\operatorname{B}_{p}(\xi,\eta)>0\qquad\Longleftrightarrow\qquad
p\not\in(\xi,\eta).$

The second fact is a consequence of the topological transitivity of the
geodesic flow on the unit tangent bundle to the surface. It can be used to
show that for any $\xi\neq\eta$ in $\partial\Gamma$ there exists an infinite
sequence $\\{\gamma_{n}\\}$ in $\Gamma$ so that

$\xi=\lim_{n\to\infty}\gamma_{n}^{+},\qquad\eta=\lim_{n\to\infty}\gamma_{n}^{-}$

where $\gamma_{n}^{-},\gamma_{n}^{+}\in\partial\Gamma$ denote the repelling
and the attracting points of $\gamma_{n}\in\Gamma$.

With these observations we can proceed to the proof of Theorem A. Using
Proposition 3.2, let us pick $(\xi_{1},\xi_{4})$ and $(\xi_{2},\xi_{3})$ that
are unlinked and aligned in $\partial\Gamma$ while
$(\phi(\xi_{1}),\phi(\xi_{4}))$ and $(\phi(\xi_{2}),\phi(\xi_{3}))$ are linked
in a copy of a hyperbolic plane $\partial{\mathbf{H}}^{2}$ contained in the
hyperbolic space ${\mathbf{H}}^{3}$. Let $p\in{\mathbf{H}}^{3}$ denote the
intersection of the linked geodesic lines $(\phi(\xi_{1}),\phi(\xi_{4}))$ and
$(\phi(\xi_{2}),\phi(\xi_{3}))$. Since these two geodesic lines are distinct,
$p\not\in(\phi(\xi_{2}),\phi(\xi_{4}))$, and therefore, using the first fact,
we obtain

$\delta=\operatorname{B}_{p}(\phi(\xi_{2}),\phi(\xi_{4}))>0.$

We can now use the second fact, and find sequences $\\{a_{n}\\}$ and
$\\{b_{n}\\}$ in $\Gamma$, so that

$a_{n}^{-}{\buildrel\over{\longrightarrow}}\xi_{1},\qquad
a_{n}^{+}{\buildrel\over{\longrightarrow}}\xi_{4},\qquad
b_{n}^{-}{\buildrel\over{\longrightarrow}}\xi_{2},\qquad
b_{n}^{+}{\buildrel\over{\longrightarrow}}\xi_{3}.$

Denote $A_{n}=\pi(a_{n})$ and $B_{n}=\pi(b_{n})$ the corresponding elements in
$\operatorname{PSL}_{2}({\mathbf{C}})$. Note that
$\phi(a_{n}^{\pm})=A_{n}^{\pm}$ and $\phi(b_{n}^{\pm})=B_{n}^{\pm}$ are the
repelling/attracting points in $\partial{\mathbf{H}}^{3}$. Upon replacing
$a_{n},b_{n}$ by their powers, we may assume that

$\ell_{{\mathbf{H}}^{3}}(A_{n}){\buildrel\over{\longrightarrow}}\infty,\qquad\ell_{{\mathbf{H}}^{3}}(B_{n}){\buildrel\over{\longrightarrow}}\infty.$

Let $p_{n}^{A}$ denote the projection of point $p$ to the geodesic line
$(\phi(a_{n}^{-}),\phi(a_{n}^{+}))$ which is the axis
$\operatorname{Ax}_{{A_{n}}}$ in ${\mathbf{H}}^{3}$. Since
$\phi:\partial\Gamma{\buildrel\over{\longrightarrow}}\partial{\mathbf{H}}^{3}$
is continuous,
$A_{n}^{-}=\phi(a_{n}^{-}){\buildrel\over{\longrightarrow}}\phi(\xi_{1})$ and
$A_{n}^{+}=\phi(a_{n}^{+}){\buildrel\over{\longrightarrow}}\phi(\xi_{4})$.
This implies

$d_{{\mathbf{H}}^{3}}(p_{n}^{A},p){\buildrel\over{\longrightarrow}}0.$

Similarly, denoting by $p_{n}^{B}\in{\mathbf{H}}^{3}$ the projection of $p$ to
the geodesic line $(\phi(\xi_{2}),\phi(\xi_{3}))$ which is the axis
$\operatorname{Ax}_{{B_{n}}}$ in ${\mathbf{H}}^{3}$, we get
$d_{{\mathbf{H}}^{3}}(p_{n}^{B},p){\buildrel\over{\longrightarrow}}0$.

$B_{n}^{-}$$B_{n}^{+}$$A_{n}^{+}$$A_{n}^{-}$$\phi(\xi_{4})$$\phi(\xi_{1})$$\phi(\xi_{2})$$\phi(\xi_{3})$$p$$p^{B}_{n}$$x_{n}=B_{n}^{-1}.p^{B}_{n}$$p_{n}^{A}$$y_{n}=A_{n}.p^{A}_{n}$

Now consider the points $x_{n}=B_{n}^{-1}.p_{n}^{B}$ and
$y_{n}=A_{n}.p_{n}^{A}$. Since $p_{n}^{b}=B^{n}.x_{n}$ and $x_{n}$ are on the
axis $\operatorname{Ax}_{{B_{n}}}$ of $B_{n}$ we have
$d_{{\mathbf{H}}^{3}}(p_{n}^{b},x_{n})=\ell_{{\mathbf{H}}^{3}}(B_{n})$ and

(4.1) $|d_{{\mathbf{H}}^{3}}(p,x_{n})-\ell_{{\mathbf{H}}^{3}}(B_{n})|\leq
d_{{\mathbf{H}}^{3}}(p,p_{n}^{B}){\buildrel\over{\longrightarrow}}0.$

Similarly,

(4.2) $|d_{{\mathbf{H}}^{3}}(p,y_{n})-\ell_{{\mathbf{H}}^{3}}(A_{n})|\leq
d_{{\mathbf{H}}^{3}}(p,p_{n}^{A}){\buildrel\over{\longrightarrow}}0.$

Hence

$\lim_{n\to\infty}x_{n}=\lim_{n\to\infty}A_{n}^{+}=\phi(\xi_{2}),\qquad\lim_{n\to\infty}y_{n}=\lim_{n\to\infty}B_{n}^{-}=\phi(\xi_{4}).$

Therefore

(4.3)
$\lim_{n\to\infty}\left(d_{{\mathbf{H}}^{3}}(x_{n},p)+d_{{\mathbf{H}}^{3}}(p,y_{n})-d_{{\mathbf{H}}^{3}}(x_{n},y_{n})\right)=\operatorname{B}_{p}(\phi(\xi_{2}),\phi(\xi_{4}))=\delta>0.$

We also have

$\begin{split}d_{{\mathbf{H}}^{3}}\left((A_{n}B_{n}).x_{n},y_{n}\right)&=d_{{\mathbf{H}}^{3}}(A_{n}.p_{n}^{B},y_{n})\\\
&=d_{{\mathbf{H}}^{3}}(A_{n}.p_{n}^{B},A_{n}.p_{n}^{A})=d_{{\mathbf{H}}^{3}}(p_{n}^{B},p_{n}^{A})\\\
&\leq
d_{{\mathbf{H}}^{3}}(p_{n}^{B},p)+d_{{\mathbf{H}}^{3}}(p,p_{n}^{A}){\buildrel\over{\longrightarrow}}0.\end{split}$

Using (4.1), (4.2), (4.3) we deduce

$\lim_{n\to\infty}\left(\ell_{{\mathbf{H}}^{3}}(A_{n})+\ell_{{\mathbf{H}}^{3}}(B_{n})-d_{{\mathbf{H}}^{3}}(A_{n}B_{n}.x_{n},x_{n})\right)=\delta.$

Since $\ell_{{\mathbf{H}}^{3}}(A_{n}B_{n})\leq
d_{{\mathbf{H}}^{3}}(A_{n}B_{n}.x_{n},x_{n})$, it follows that

$\liminf_{n\to\infty}\left(\ell_{{\mathbf{H}}^{3}}(A_{n})+\ell_{{\mathbf{H}}^{3}}(B_{n})-\ell_{{\mathbf{H}}^{3}}(A_{n}B_{n})\right)\geq\delta.$

The latter fact can be rewritten as

$\liminf_{n\to\infty}\left(\ell_{q}(a_{n})+\ell_{q}(b_{n})-\ell_{q}(a_{n}b_{n})\right)\geq\delta.$

Recall that $(\xi_{1},\xi_{4})$ and $(\xi_{2},\xi_{3})$ are unlinked and
aligned in $\partial\Gamma$, and are approximated by $(a_{n}^{-},a_{n}^{+})$
and $(b_{n}^{-},b_{n}^{+})$ respectively. Thus, we can find $k\in{\mathbf{N}}$
large enough, so that the pair of elements $a=a_{k}$, $b=b_{k}$ satisfy

$\ell_{q}(a)+\ell_{q}(b)-\ell_{q}(ab)>\frac{1}{2}\delta$

while $(a^{-},a^{+})$ and $(b^{-},b^{+})$ are unlinked and aligned. By Theorem
2.1 the latter condition implies that for every $g\in\mathscr{R}(\Sigma)$ we
have

$\ell_{g}(a)+\ell_{g}(b)-\ell_{g}(ab)<0.$

Thus

$\frac{\ell_{q}(a)+\ell_{q}(b)}{\ell_{q}(ab)}:\frac{\ell_{g}(a)+\ell_{g}(b)}{\ell_{g}(ab)}>\frac{\ell_{q}(a)+\ell_{q}(b)}{\ell_{q}(ab)}>1+\frac{\delta}{2\ell_{q}(ab)}.$

We deduce that for every $g\in\mathscr{R}(\Sigma)$ we get

$\rho_{\operatorname{Lip}}(\delta_{q},\delta_{g})>\log(1+\frac{\delta}{2\ell_{q}(ab)})>0.$

This completes the proof of Theorem A.

## References