and thus

\begin{align*}
N_\pi(T)= & N_\pi(t_0)
\\
= & N_\pi(t_0-1)+1
\\
\leq & \psi(t_0)+1
\\
\leq & \psi(T)+1 .
\end{align*}

Next, for each timestep $t$, we bound the estimation error of $f(\bw)$.
For any $t>0$, let $\Delta_t \triangleq f(\bw^*)-f(\bw_t)$ and $\hat{f}_{t}(\bw) \triangleq \bw^\top \bs{\hat{\theta}}_{t} - \rho \bw^\top \hat{\Sigma}_{t} \bw$.
Suppose that event $\cM_t \cap \cN_t$ occurs. Then, according to Lemmas <ref> and <ref>, we have that for any $\bw \in \triangle_{d}$,

\begin{align*}
\abr{ f(\bw) - \hat{f}_{t-1}(\bw)  } \leq & \sqrt{2\beta(\delta_t)} \sqrt{\bw^\top B_\pi^+ D_{t-1}^{-1} \sbr{\lambda \Lambda_{\Sigma_\pi} D_{t-1} + \sum_{s=1}^{N_\pi(t-1)} \Sigma_\pi}  D_{t-1}^{-1} (B_\pi^{+}) \top \bw} \\& + 5 \rho   \| C^+_\pi \|  \sqrt{\frac{ 3 \ln t}{2 N_\pi(t-1)}}
\\
\leq & \sqrt{ 2 \ln(t \ln^2 t) + \ln \ln t + \frac{\tilde{d}}{2} \ln(1+\frac{e}{\lambda}) } \cdot \sqrt{ \bw^\top B_\pi^+ D_{t-1}^{-1} \sbr{\lambda \Lambda_{\Sigma_\pi} + \Sigma_\pi}  (B_\pi^{+}) \top \bw  } \\& + 5 \rho   \| C^+_\pi \|  \sqrt{\frac{3 \ln t}{2 N_\pi(t-1)}}
\\
\leq & 7  \sqrt{  \ln t + \frac{\tilde{d}}{2} \ln(1+\frac{e}{\lambda}) } \cdot \sbr{ \sqrt{ \bw^\top B_\pi^+  \sbr{\lambda \Lambda_{\Sigma_\pi} + \Sigma_\pi}  (B_\pi^{+}) \top \bw  } + \rho \| C^+_\pi \| } \cdot \sqrt{\frac{1}{ N_\pi(t-1)}}
\end{align*}

Let $Z(\rho, \pi) \triangleq \max_{\bw \in \triangle_{d}} \sbr{ \sqrt{ \bw^\top B_\pi^+  \sbr{\lambda \Lambda_{\Sigma_\pi} + \Sigma_\pi}  (B_\pi^{+}) \top \bw  } + \rho \| C^+_\pi \| }$. Then, we have

\begin{align*}
\abr{ f(\bw) - \hat{f}_{t-1}(\bw)  }
\leq  7  \cdot Z(\rho, \pi)  \sqrt{ \frac{\ln t + \tilde{d} \ln(1+e/\lambda) }{ N_\pi(t-1)} } .
\end{align*}

Let $\cL^{\textup{exploit}}(t)$ denote the event that algorithm $\algfullbandit$ does the exploitation at timestep $t$.
For any $t>0$, if $\cL^{\textup{exploit}}(t)$ occurs, we have $N_\pi(t-1)>\psi(t)$. Thus

\begin{align*}
\abr{ f(\bw) - \hat{f}_{t-1}(\bw)  }
< & 7  \cdot Z(\rho, \pi)  \sqrt{ \frac{\ln t + \tilde{d} \ln(1+e/\lambda)}{ \psi(t) } }
\\
= & 7  \cdot Z(\rho, \pi)  \sqrt{ d \sbr{\ln t + \tilde{d} \ln(1+e/\lambda)} } \cdot t^{-\frac{1}{3}}
\end{align*}

According to Lemmas <ref>,<ref>, for any $t \geq 2$, we bound the probability of event $\neg(\cM_t \cap \cN_t)$ by

\begin{align*}
\Pr \mbr{ \neg(\cM_t \cap \cN_t) } \leq &  \frac{6 d^2}{t^2} + \frac{1}{t \ln^2 t}
\\
\leq &  \frac{6 d^2}{t \ln^2 t} + \frac{1}{t \ln^2 t}
\\
= &  \frac{7 d^2}{t \ln^2 t}
\end{align*}

The expected regret of algorithm $\algfullbandit$ can be divided into two parts, one  due to exploration and the other due to exploitation. Then, we can obtain

\begin{align*}
\ex[\cR(T)] \leq & N_\pi(T) \cdot \tilde{d} \Delta_{\textup{max}} + \sum_{t=1}^T \ex[ \Delta_t | \cL^{\textup{exploit}}(t)]
\\
\leq & (\psi(T)+1) \cdot \tilde{d} \Delta_{\textup{max}} + \sum_{t=1}^T \sbr{ \ex[ \Delta_t | \cL^{\textup{exploit}}(t), \cM_t \cap \cN_t ] + \ex[ \Delta_t | \cL^{\textup{exploit}}(t),  \neg(\cM_t \cap \cN_t)] \cdot \Pr \mbr{ \neg(\cM_t \cap \cN_t) } }
\\
= & O \sbr{\sbr{ \frac{T^{\frac{2}{3}}}{d} + 1} \cdot \tilde{d} \Delta_{\textup{max}} + \sum_{t=1}^T \sbr{  Z(\rho, \pi)  \sqrt{ d(\ln t + \tilde{d}\ln(1+e/\lambda) )} \cdot t^{-\frac{1}{3}}  + \Delta_{\textup{max}} \cdot  \frac{d^2}{t \ln^2 t}  } }
\\
= & O \sbr{T^{\frac{2}{3}} d \Delta_{\textup{max}} + d^2 \Delta_{\textup{max}} +  Z(\rho, \pi)  \sqrt{ d(\ln T + d^2 \ln(1+\lambda^{-1}))} \cdot  T^{\frac{2}{3}}  +  d^2 \Delta_{\textup{max}}}
\\
= & O \sbr{ Z(\rho, \pi)  \sqrt{ d(\ln T + d^2 \ln(1+\lambda^{-1})) } \cdot  T^{\frac{2}{3}}+  d \Delta_{\textup{max}} \cdot  T^{\frac{2}{3}} + d^2 \Delta_{\textup{max}} }
\\
= & O \sbr{ Z(\rho, \pi)  \sqrt{ d(\ln T + d^2 \ln(1+\lambda^{-1})) } \cdot  T^{\frac{2}{3}}+   d \Delta_{\textup{max}} \cdot  T^{\frac{2}{3}} }
\end{align*}

Choosing $\lambda=\frac{1}{2}$ and using $\Lambda_{\Sigma_\pi}=\Sigma_\pi$, we obtain

\begin{align*}
\ex[\cR(T)]  =  O \sbr{ Z(\rho, \pi)  \sqrt{ d(\ln T + d^2 ) } \cdot  T^{\frac{2}{3}}+   d \Delta_{\textup{max}} \cdot  T^{\frac{2}{3}} },
\end{align*}

where $Z(\rho, \pi) = \max_{\bw \in \triangle_{d}} \sbr{ \sqrt{ \bw^\top B_\pi^+  \Sigma_\pi  (B_\pi^{+}) \top \bw  } + \rho \| C^+_\pi \| }$.


#  A detailed study on novel metamaterial absorber at WR3 band

Sarvesh Gharat  Department of Electronics and Telecommunication
Vishwakarma Institute of Information Technology
Pune, India
<EMAIL_ADDRESS>Shriganesh Prabhu  Department of Condensed Matter
Physics
Tata Institute of Fundamental Research
Mumbai, India
<EMAIL_ADDRESS>
###### Abstract

Frequency band of 30 to 300 GHz, which is termed as mm wave band offers 100
times bandwidth as compared to sub 6 GHz band. In this study, we propose a
novel design which acts as a near unity absorber at 258 GHz or 0.258 THz in Y
polarisation and 292 GHz or 0.292 THz in X polarisation. The study also
includes effect of change in parameters on resonance frequency in both X and Y
polarisation which allows us to use same design with modified parameters to
get near unity absorptivity in frequency of 205 to 345 GHz. Further, we also
discuss on higher frequency mode which particularly appears in Y polarisation
and on changing the inner square dimension of the design.

###### Keywords:

Absorber, Kapton, Metamaterials, Terahertz, 6G

##  I Introduction

There are some properties such as negative permitivity and permeability which
are absent in materials which are present in nature. Introducing metamaterials
helps us to achieve all these properties [1][2][3]. This unique properties of
metamaterials are mainly observed at it’s resonant frequencies [4].

Metamaterials can be treated as an LC circuit. Both inductance and capacitance
depends on structure of metamaterial, polarisation of incident wave, thickness
of substrate and refractive index of substrate used [5][6][7]. Although, we
are familiar with variation of these properties for specific structures
[8][9][10][11], we don’t have any universal theoretical model helping us to
get the exact relations.

Use of metamaterials is widely done at many places such as sensor
identification, high-frequency battlefield communication, frequency filter and
many more [12][13][14]. One such application is using it as a perfect unity
absorber. These metamarial absorbers are further widely used is applications
such as thermal emitters, photovoltaic cells, optical imaging devices, etc
[15] [16] [17]. The first metamaterial absorber was experimentally confirmed
by Landy et al. [18] in 2008. Since than, a lot of work has been done in
creating novel metamaterial absorbers [19][20][21][22][23] to be used in
different applications as stated before.

In today’s era where telecommunication industry is developing exponentially,
it is important to also have near unity absorbers in different bands of 6G
[24][25]. In our previous work [26], we had propose a novel metamaterial
absorber in WR 1 and WR 1.5 bands. However, there is a strong need to have
near unity absorbers in remaining bands of 6G. Hence, in this study we propose
a novel metamaterial structure which will act as a near unity absorber in
specific frequencies of WR 3 band. Along with that, we also provide a study on
change in central frequencies due to variation in Geometrical parameters and
change in polarisation.

##  II Design and Simulations

This section focuses on schematic of proposed design. Along with that, this
section will also includes the necessary boundary conditions and assumptions
considered in this study.

Figure 1: Proposed Design

The proposed design as shown in Figure 1 has symmetry along both X and Y axis.
The proposed design is simulated over Kapton with thickness of 25$\mu$m having
refractive index of $1.88+0.04i$.

Parameter | Value
---|---
p | 300
a | 202.5
c | 224
s | 90
g | 22.5
t | 15
TABLE I: Value of Parameters in $\mu$m as shown in Figure1

All the values of parameters as shown in Figure1 can be seen in TableI. The
parameters given in TableI happen to be optimized parameters to get resonant
frequency of 235 GHz.

To simulate the proposed design, use of COMSOL MULTIPHYSICS ®, is done. Utmost
care is taken while considering boundary conditions so as to reduce the
required computational power. The used conditions are elaborated below:

  * •

Perfectly Electric Conductor: It is a special case of electric field boundary
condition that sets tangential component of electric field to zero. It is
generally used in modelling a lossless metallic surface [27].

  * •

Periodic Condition: Periodic condition is used to make unit cell structure
periodic along particular direction [27].

In the proposed simulation, the structure is periodic along both X and Y
direction. Along with that, use of Perfectly Electric Condition (PEC)
condition is done on the design as well as the end of the substrate. Adding
PEC boundary condition at the bottom of the substrate allows us to omit air
domain after substrate helping us to reduce the computational power.

To vary the parameters as listed in TableI, use of parametric sweep is done.
The parametric sweep feature as provided by COMSOL[27] helps us to iterate the
process of finding a solution over complete set of variable. This, not only
helps us in reducing the manual work but also helps in comparing the results
at one place. TableII tells us about the range in which individual parameters
are varied.

Parameter | Minimum Value | Maximum Value
---|---|---
a | 160 | 260
c | 180 | 280
s | 40 | 140
g | 5 | 50
TABLE II: Minimum and Maximum values of parameter

Further to calculate absorbance in percentage, we make use of simple
formulation between absorbance, transmission and reflection which is

$\text{i.e Absorptivity }=1-\text{abs}(S_{11})^{2}-\text{abs}(S_{21})^{2}$

where $S_{11}$ and $S_{21}$ are reflection and transmission coefficients.
However, as in simulation we make use of PEC at the bottom, the incident wave
cant pass through it making $S_{21}$ as 0 and simplifying the formula to

$\text{i.e Absorptivity in }\%=1-\text{abs}(S_{11})^{2}$

The proposed design is studied in both X and Y polarisation. Further, all the
data is extracted in a text file which is further plotted using Origin Pro
2021.

##  III Results and Discussion

This section comprises of simulated optimised results. Apart from that we will
be also discussing on effects of geometrical parameters on Absorbance in both
X and Y polarisation.

###  III-A Y Polarisation

####  III-A1 Optimized Parameters

As seen in Figure 2, almost 96.3% of incident EM field is absorbed at
frequency of 258 GHz or 0.258 THz. Apart from that there’s one more small
resonance happening at 0.536 THz with absorptivity of 35.12%.

Figure 2: Frequency vs Absorbance plot for Y polarisation

The initial low frequency mode as seen in Figure 2 is mainly due to inductance
produced after the confinement of electric field on horizontal arms as shown
in Figure 3

Figure 3: $E_{z}$ component of Electric field on metamaterial at low frequency
mode

The supporting fact for above happening is evident from Figure 4 where we have
shown surface current density.

Figure 4: Surface Current Density on metamaterial at low frequency mode in Y
polarisation

We see that the current follows a unidirectional path along Y axes. This is
consistent with polarisation of Electric field. Apart from that, we also see
high current density is both inner and outer vertical arms which leads to
attenuation of emitted radiation causing a very high absorbance at low
frequency mode.

On contrary to that the higher frequency mode appears due to creation of
capacitance between inner and outer horizontal arms as seen in Figure 5.

Figure 5: $E_{z}$ component of Electric field on metamaterial at high
frequency mode in Y polarisation

Also in this case, there is flip in direction of current in all vertical arms.
This happens to be in opposite direction of the polarisation. With that, the
current density is comparatively less than that of what we saw in initial
lower frequency mode. This can be concluded by looking into Figure 4 and
Figure 6

Figure 6: Surface Current Density on metamaterial at high frequency mode in Y
polarisation

####  III-A2 Variation in Parameters

To study effect of Geometrical parameters like height (c), width (a), gap (g)
and size of square (s) as seen in Figure 1, we vary them with minimum and
maximum values as shown in Table II

Figure 7: Contour plot of absorbance on varying height in Y polarisation

Figure 7 refers to variation of resonant frequency on changing height. We see
that on increasing the height, the initial mode frequency gets redshifted. As
the period remains constant and the height increases, the gap between 2 unit
cells decrease resulting into more capacitance and as we know, more is the
capacitance, less wil be the resonant frequency. On the other hand, in higher
mode, along with redshift we also see decrease in absorbance and at one point
the peak vanishes. This happens because the gap between the upper two parallel
arms increases as the length is increased and this reduces the mode that can
be sustained within the gap between the two arms. Thus at one point it can
become weak enough to vanish.

Figure 8: Contour plot of absorbance on varying width in Y polarisation

The redshift is also observed on increasing the width as seen in Figure 8.
However, in this case the reason for this happening is increase in inductance
due to increase in horizontal arm length. On contrary to Figure 7, in Figure 8
we see that higher mode vanishes on increasing the width. This can happen as
the width is increased, the two side of the cell will start coming closer and
then there is a coupling between the two progressively increasing as the width
increases. This can result in the appearance of the mode, which is
capacitively excited and as the arms draw closer, the capacitance decreases,
increasing the frequency. This the mode appears suddenly as the width becomes
larger and the mode becomes stronger.

Figure 9: Contour plot of absorbance on varying gap in Y polarisation

We further vary gap to understand confirm the production of capacitance in the
gap. However, if that was the case then there would had been blueshift in
resonant frequency due to decrease in capacitance. But there’s no such
observation in the initial mode (See Figure 9, hence we can safely say that
there’s negligible amount of capacitance created in first mode. It is obvious
from the figure that when gap will vary, the length remains the same, so
frequency should not change. This can also be verified from Figure 3 where we
don’t see any confinement of electric field. On other hand, in high frequency
mode, a considerable amount of capacitance is been created as seen in Figure
5. Hence, there’s small amount of redshift happening at higher frequency as
evident from Figure 9

Figure 10: Contour plot of absorbance on varying size of square in Y
polarisation

From Figure 10, we observe that similar to gap, the internal square plays no
major role in initial mode. This is expected as the first mode is due to the
upper and lower arm gap capacitance. However, at high frequency, on increasing
the inner square size, mode gets redshifted before vanishing away. However it
reappears after getting close to exterior loop. This will happen if the inner
square , after increasing it’s size can start interacting with the upper and
lower arm excitation mode. The increasing size of the central portion can
result in the modified coupling of the upper-lower arms. So the Frequency
start reducing and will vanish at a particular combination of Size and arms
length. Now if the size of the inner square increases further, now the
capacitance between the upper and lower arms will start dominating, and there
will be arise of new frequency modes.Other way to look at it is, the
capacitance start increasing with size, decreasing the resonance frequency of
the high frequency mode.

###  III-B X Polarisation

####  III-B1 Optimized Parameters

From Figure 11, we see that on exciting electric field in X direction, there
only one mode occurring in bandwidth of 500 GHz at 292 GHz with absorptivity
of 93.85 %.

Figure 11: Frequency vs Absorbance plot for X polarisation

In this case, the only mode visible is majorly due to the inductance produce
in vertical arms as seen in Figure 12.

Figure 12: $E_{z}$ component of Electric field on metamaterial at low
frequency mode in X polarisation

The surface current density as observed in Figure 13 is unidirectional towards
the direction of polarisation. With that, the surface charge density is also
negligible in inner square. Apart from that the major reason behind high
absorptivity at this frequency is due to attenuation of incident wave because
of high current density as seen in Figure 13

Figure 13: Surface Current Density on metamaterial at low frequency mode in X
polarisation

####  III-B2 Variation in Parameters

We further vary the geometrical parameters as shown in Table II to study
effect of these parameters on absorbance.

Figure 14: Contour plot of absorbance on varying height in X polarisation

Figure 14 refers to change in absorptivity on varying the height. For all the
values of height considered in this study we have one single mode in frequency
from 0.1 to 0.55 THz. However, as we move towards higher frequency there are
some instances of higher frequency mode which currently isn’t our interest. We
also observe a red shift is resonance frequency. This is due to increase in
inductance on increasing the arm length.

Figure 15: Contour plot of absorbance on varying width in X polarisation

Similarly, on increasing the width we observe a redshift. Here, the redshift
occurs mainly because of increase in capacitance which happens due to
closeness of exterior arms in every unit cell. Apart from that as in X
polarisation, the initial gap doesn’t play any role there is no instance of
second mode in this case. Th supporting instance for this claim is evident
from Figure 13 and Figure 16.

Figure 16: Contour plot of absorbance on varying gap in X polarisation

In Figure 16, we see that there’s no change in resonance frequency on changing
the gap size. This happens due to direction of Electric field being parallel
to gap. As both the ends of gap will be on equipotential surface in case of X
polarisation, theres no capacitance created in this case resulting into no
effect of absorbance due to gap.

Figure 17: Contour plot of absorbance on varying size of square in X
polarisation

Similar thing can be observed on changing the size of square as shown in
Figure 17, However, in this case we also see a higher mode which starts
appearing from 0.5 THz. The mode exists at the higher frequency, but is not
visible in our simulations data. However, as the middle square changes it’s
size, the capacitive coupling must be getting reduced, increasing the
capacitance, decreasing the resonance frequency. This trend is clearly seen in
the figure, where the second mode keeps red shifting, eventually vanishing
when the coupling is no more effective.

##  IV Conclusion

In this study, we propose a novel metamaterial structure to be used in 6G
band. This design happens to be one of the optimal candidate for 6G
applications mainly due to it’s low complexity. We further provide a study on
effect on resonance frequency on changing of geometric parameters like width,
height, gap and size of square in both X and Y polarisation. The results as
achieved on changing the dimensions tell us that the same design with some
more optimal dimensions can also be used in multiple applications,.

## References

  * [1] R. S. Kshetrimayum, “A brief intro to metamaterials,” _IEEE potentials_ , vol. 23, no. 5, pp. 44–46, 2004.
  * [2] F. Capolino, _Theory and phenomena of metamaterials_. CRC press, 2017.
  * [3] R. M. Walser, “Metamaterials: What are they? what are they good for?” in _APS March Meeting Abstracts_ , 2000, pp. Z5–001.
  * [4] S. S. Bukhari, J. Y. Vardaxoglou, and W. Whittow, “A metasurfaces review: Definitions and applications,” _Applied Sciences_ , vol. 9, no. 13, p. 2727, 2019.
  * [5] B. R. Sangala, A. Nagarajan, P. Deshmukh, H. Surdi, G. Rana, V. G. Achanta, and S. Prabhu, “Single and multiband thz metamaterial polarisers,” _Pramana_ , vol. 94, no. 1, pp. 1–6, 2020.
  * [6] D. R. Chowdhury, R. Singh, M. Reiten, H.-T. Chen, A. J. Taylor, J. F. O’Hara, and A. K. Azad, “A broadband planar terahertz metamaterial with nested structure,” _Optics express_ , vol. 19, no. 17, pp. 15 817–15 823, 2011\.
  * [7] H. Tao, C. Bingham, A. Strikwerda, D. Pilon, D. Shrekenhamer, N. Landy, K. Fan, X. Zhang, W. Padilla, and R. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” _physical review B_ , vol. 78, no. 24, p. 241103, 2008\.
  * [8] A. Marwaha _et al._ , “An accurate approach of mathematical modeling of srr and sr for metamaterials.” _Journal of Engineering Science & Technology Review_, vol. 9, no. 6, 2016.
  * [9] B. George, B. Nair, and S. K. Menon, “Mathematical modeling and validation of a hexagonal split ring resonator,” in _2018 Second International Conference on Advances in Electronics, Computers and Communications (ICAECC)_. IEEE, 2018, pp. 1–4.
  * [10] V. E. Elander, “Mathematical modeling of metamaterials,” 2011.
  * [11] S. Bose, M. Ramaraj, S. Raghavan, and S. Kumar, “Mathematical modeling, equivalent circuit analysis and genetic algorithm optimization of an n-sided regular polygon split ring resonator (nrpsrr),” _Procedia Technology_ , vol. 6, pp. 763–770, 2012.
  * [12] A. Valipour, M. H. Kargozarfard, M. Rakhshi, A. Yaghootian, and H. M. Sedighi, “Metamaterials and their applications: an overview,” _Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications_ , p. 1464420721995858, 2021.
  * [13] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal, A. Alkhateeb, and G. C. Trichopoulos, “Wireless communications and applications above 100 ghz: Opportunities and challenges for 6g and beyond,” _IEEE access_ , vol. 7, pp. 78 729–78 757, 2019.
  * [14] G. Singh and A. Marwaha, “A review of metamaterials and its applications,” 2015\.
  * [15] M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” _Physical Review B_ , vol. 79, no. 3, p. 033101, 2009.
  * [16] J. N. Munday and H. A. Atwater, “Large integrated absorption enhancement in plasmonic solar cells by combining metallic gratings and antireflection coatings,” _Nano letters_ , vol. 11, no. 6, pp. 2195–2201, 2011.
  * [17] H. Hu, C. Ma, and Z. Liu, “Plasmonic dark field microscopy,” _Applied Physics Letters_ , vol. 96, no. 11, p. 113107, 2010.
  * [18] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” _Physical review letters_ , vol. 100, no. 20, p. 207402, 2008.
  * [19] B.-X. Wang, Y. He, P. Lou, N. Xu, X. Wang, Y. Wang, and J. Cao, “Multiple-band terahertz metamaterial absorber using multiple separated sections of metallic rectangular patch,” _Frontiers in Physics_ , vol. 8, p. 308, 2020.
  * [20] C. Gandhi, P. R. Babu, and K. Senthilnathan, “Ultra-thin polarization independent broadband terahertz metamaterial absorber,” _Frontiers of Optoelectronics_ , vol. 14, no. 3, pp. 288–297, 2021.
  * [21] Z. Liu, X. Han, and A. Wang, “Ultrathin polarization-insensitive tri-band thz perfect metamaterial absorber,” _EPJ Applied Metamaterials_ , vol. 7, p. 2, 2020.
  * [22] Z. Luo, S. Ji, J. Zhao, H. Wu, and H. Dai, “A multiband metamaterial absorber for ghz and thz simultaneously,” _Results in Physics_ , vol. 30, p. 104893, 2021.
  * [23] A. Elakkiya, R. Sankararajan, and B. Sreeja, “Seven-band ultra-thin terahertz metamaterial absorber at 0.3-0.5 thz frequency,” _Circuit World_ , 2020.
  * [24] D. T. Chi, B. X. Khuyen, B. S. Tung, V. D. Lam, L. Y. Chen, and Y. Lee, “Progresses in metamaterials for advanced low-frequency perfect absorbers: a brief review,” _Journal of Electromagnetic Waves and Applications_ , vol. 34, no. 17, pp. 2251–2265, 2020.
  * [25] Z. Li, J. Pan, H. Hu, and H. Zhu, “Recent advances in new materials for 6g communications,” _Advanced Electronic Materials_ , p. 2100978, 2021.
  * [26] S. Gharat, P. Kulkarni, S. Prabhu, and C. Garde, “Dualband and tripleband metamaterial absorber in wr1 band and lower terahertz frequencies,” 2022. [Online]. Available: https://arxiv.org/abs/2203.13442
  * [27] C. Multiphysics, “Introduction to comsol multiphysics®,” _COMSOL Multiphysics, Burlington, MA, accessed Feb_ , vol. 9, p. 2018, 1998\.


# On conciseness of the word in Olshanskii’s example

Matteo Pintonello  Matteo Pintonello: Department of Mathematics, Euskal
Herriko Unibertsitatea UPV/EHU, 48940 Leioa, Spain<EMAIL_ADDRESS>and  Pavel Shumyatsky  Pavel Shumyatsky: Department of Mathematics, University
of Brasilia, Brasilia DF, Brazil<EMAIL_ADDRESS>
###### Abstract.

A group-word $w$ is called concise if the verbal subgroup $w(G)$ is finite
whenever $w$ takes only finitely many values in a group $G$. It is known that
there are words that are not concise. In particular, Olshanskii gave an
example of such a word, which we denote by $w_{o}$. The problem whether every
word is concise in the class of residually finite groups remains wide open. In
this note we observe that $w_{o}$ is concise in residually finite groups.
Moreover, we show that $w_{o}$ is strongly concise in profinite groups, that
is, $w_{o}(G)$ is finite whenever $G$ is a profinite group in which $w_{o}$
takes less than $2^{\aleph_{0}}$ values.

###### Key words and phrases:

Conciseness, strong conciseness, profinite groups, word problems

###### 2010 Mathematics Subject Classification:

Primary 20F10

This paper was written while the second author was visiting Department of
Mathematics of the University of the Basque Country. He expresses his sincere
gratitude to the department for excellent hospitality. The first author was
supported by the Spanish Government project PID2020-117281GB-I00, partially by
FEDER funds, by the Basque Government project IT483-22, and grant FPI-2018 of
the Spanish Government. He is extremely thankful to G. Fernández-Alcober and
M. Casals for the useful suggestions.

##  1\. Introduction

A group-word $w$ is called concise in the class of groups $\mathcal{C}$ if the
verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values
in a group $G\in\mathcal{C}$. In the sixties Hall raised the problem whether
every word is concise in the class of all groups but in 1989 S. Ivanov [13]
solved the problem in the negative. More precisely, he showed that if $n\geq
10^{10}$ is an odd integer and $p\geq 5000$ is a prime, there is a torsion-
free group in which the word $[[x^{pn},y^{pn}]^{n},y^{pn}]^{n}$ takes only two
values.

Another noteworthy word was introduced by Olshanskii in [17]. For positive
integers $d$ and $n$, set

$v(x,y)=[[x^{d},y^{d}]^{d},[y^{d},x^{-d}]^{d}]$

and

$w_{o}(x,y)=[x,y]v(x,y)^{n}[x,y]^{\varepsilon_{1}}v(x,y)^{n+1}\cdots[x,y]^{\varepsilon_{h-1}}v(x,y)^{n+h-1},$

where

$\varepsilon_{10k+1}=\varepsilon_{10k+2}=\varepsilon_{10k+3}=\varepsilon_{10k+5}=\varepsilon_{10k+6}=1$
$\varepsilon_{10k+4}=\varepsilon_{10k+7}=\varepsilon_{10k+8}=\varepsilon_{10k+9}=\varepsilon_{10k+10}=-1$

for $k=0,1,\ldots,(h-1)/10$ and $h\equiv 1\mod 10$, $h>50000$.

Olshanskii showed that the parameters $n$ and $d$ can be chosen in such a way
that the word $w_{o}$ has several remarkable properties. In particular, the
variety of groups where $w_{o}$ is a law contains infinite non-abelian groups
while all finite groups in the variety are abelian. Moreover, the word is not
concise in the class of all groups (see [18, p. 439]).

From now on, throughout the paper $w_{o}$ stands for Olshanskii’s word with
the above properties.

The problem whether all words are concise in residually finite groups remains
wide open (cf. Segal [19, p. 15] or Jaikin-Zapirain [14]). In recent years
several new positive results with respect to this problem were obtained (see
[1, 11, 9, 10, 5, 6, 7]). In the present paper we observe that the word
$w_{o}$ is concise in residually finite groups (Theorem 3.1).

A natural variation of the notion of conciseness for profinite groups was
introduced in [4]: the word $w$ is strongly concise in a class of profinite
groups $\mathcal{C}$ if the verbal subgroup $w(G)$ is finite in any group
$G\in\mathcal{C}$ in which $w$ takes less than $2^{\aleph_{0}}$ values. Here
and throughout the paper, whenever $G$ is a profinite group we write $w(G)$ to
denote the closed subgroup generated by $w$-values. A number of new results on
strong conciseness of group-words can be found in [4, 3, 2, 16, 12]. In this
note we will show that the word $w_{o}$ is strongly concise in profinite
groups (Theorem 3.4).

##  2\. Preliminaries

By a subgroup of a profinite group we always mean a closed subgroup and by a
homomorphism of profinite groups a continuous homomorphism.

######  Lemma 2.1.

[5, Lemma 4] Let $w$ be a word and $G$ a group such that the set of $w$-values
in $G$ is finite with at most $m$ elements. Then the order of $w(G)^{\prime}$
is $m$-bounded.

######  Lemma 2.2 ([4] Lemma 2.2).

Let $G$ be a profinite group and $g\in G$ be an element whose conjugacy class
$g^{G}$ contains less than $2^{\aleph_{0}}$ elements. Then $g^{G}$ is finite.

The following lemma is straightforward.

######  Lemma 2.3.

Let $v(x,y)$ be as in the introduction. In every group where $v$ is a law, and
in particular in metabelian groups, the values of the word $w_{o}$ coincide
exactly with the values of the commutator word $[x,y]$.

Using the above lemma and the fact that every finite non-abelian group
contains a non-abelian metabelian subgroup, the following result is immediate.

######  Theorem 2.4 (Lemma 29.1 of [18]).

Every finite group where $w_{o}$ is a law is abelian.

Olshanskii proved that the variety of groups satisfying the law $w_{o}\equiv
1$ contains non-abelian infinite groups. Moreover, he established the
following theorem.

######  Theorem 2.5 (Theorem 39.7 of [18]).

There is a group $G$ where $w_{o}$ takes a single nontrivial value, but
$w_{o}(G)$ is infinite.

Thus, it follows that the word $w_{o}$ is not concise.

##  3\. Main results

A group-word $w$ is called boundedly concise in the class of groups
$\mathcal{C}$ if the verbal subgroup $w(G)$ is finite of $(m,w)$-bounded order
whenever $w$ takes at most $m$ values in a group $G\in\mathcal{C}$. It is
known, due to Fernández-Alcober and Morigi [8], that every word that is
concise in the class of all groups is in fact boundedly concise. On the other
hand, it is an open problem whether the same phenomenon holds for words that
are concise in the class of residually finite groups. We can now prove the
following theorem.

######  Theorem 3.1.

The word $w_{o}$ is boundedly concise in residually finite groups.

###### Proof.

Let $m$ be a positive integer and $G$ a residually finite group in which
$w_{o}$ takes $m$ values. In view of Lemma 2.1 there is a number $f_{1}$
depending only on $m$ such that $|w_{o}(G)^{\prime}|\leq f_{1}$. If $Q$ is a
finite homomorphic image of $G$, observe that the quotient $Q/w_{o}(Q)$ is
abelian by Theorem 2.5. Hence $Q/w_{o}(Q)^{\prime}$ is metabelian. Lemma 2.3
now implies that $Q/w_{o}(Q)^{\prime}$ has at most $m$ commutators. Note that
the commutator word is boundedly concise (see for example [20] for an explicit
bound), so $|w_{o}(Q/w_{o}(Q^{\prime}))|\leq f_{2}$, where $f_{2}$ is a number
depending only on $m$. Hence $|w_{o}(Q)|\leq f_{1}f_{2}$. This holds for every
finite homomorphic image $Q$ of $G$ so we deduce that $|w_{o}(G)|\leq
f_{1}f_{2}$. ∎

We will now prove that the word $w_{o}$ is strongly concise in profinite
groups.

Start with the following lemma.

######  Lemma 3.2.

Let $G$ be a profinite group topologically isomorphic to a Cartesian product
of finite simple groups. If the word $w_{o}$ takes less than $2^{\aleph_{0}}$
values in $G$, then $G$ is finite.

###### Proof.

Write $G=\prod_{i\in I}S_{i}$, where the factors $S_{i}$ are finite simple
groups. Since every finite non-abelian group contains a non-abelian metabelian
group, it follows that $w(S_{i})$ is nontrivial for any $i\in I$. We need to
show that the index set $I$ is finite. Assume by contradiction that $I$ is
infinite. Choose a nontrivial $w_{o}$-value $c_{i}\in S_{i}$ for each $i\in
I$. Observe that for each subset $J\subseteq I$ the product $c_{J}=\prod_{i\in
J}c_{i}$ is a $w_{o}$-value. If $J_{1}\neq J_{2}$, then $c_{J_{1}}\neq
c_{J_{2}}$ and therefore $G$ contains at least $2^{\aleph_{0}}$ distinct
$w_{o}$-values, a contradiction. ∎

######  Lemma 3.3.

Let $G$ be a prosoluble group. If the word $w_{o}$ takes less than
$2^{\aleph_{0}}$ values in $G$, then the commutator subgroup $G^{\prime}$ is
finite.

###### Proof.

By Lemma 2.3, $w_{o}(G/G^{\prime\prime})=G^{\prime}/G^{\prime\prime}$.
Moreover, as the commutator word is strongly concise in profinite groups [4],
$G^{\prime}/G^{\prime\prime}$ is finite. Therefore there exists a finite set
$T$ of $w_{o}$-values such that $G^{\prime}=\langle T\rangle
G^{\prime\prime}$. Note that by Lemma 2.2 each element of $T$ has finitely
many conjugates in $G$. So we can choose $T$ in such a way that the subgroup
$\langle T\rangle$ is normal in $G$. Set $\bar{G}=G/\langle T\rangle$. Observe
that $\bar{G}$ is a prosoluble group with the property that
$\bar{G}^{\prime}=\bar{G}^{\prime\prime}$. It follows that $\bar{G}$ is
abelian and so $G^{\prime}=\langle T\rangle$.

Now by Lemma 2.2, for each $t\in T$ we have that $[\langle
T\rangle:C_{G}(t)]\leq\infty$, whence $[\langle T\rangle:Z(\langle
T\rangle)]\leq\infty$. By Schur’s Theorem, the commutator subgroup $\langle
T\rangle^{\prime}$ is finite. This implies that $G$ is finite-by-metabelian.
Factoring out $G^{\prime\prime}$ we can assume that $G$ is metabelian and
apply again Lemma 2.3. Since the commutator word is strongly concise, we
conclude that $G^{\prime}$ is finite, as required. ∎

It is well known that if $K$ is a finite group, then there exists a series

$1=K_{0}\leq K_{1}\leq\cdots\leq K_{2h+1}=K$

of normal subgroups of $K$ such that $K_{i+1}/K_{i}$ is soluble (possibly
trivial) if $i$ odd and a direct product of non-abelian simple groups if $i$
is even. The number of insoluble factors in this series is called the
insoluble length $\lambda(K)$ of $K$. Theorem 1.4 of [15] implies that if the
Sylow 2-subgroup of $K$ is soluble with derived length $l$, then $\lambda(G)$
is bounded in terms of $l$ only. We are now ready to complete the proof that
the word $w_{o}$ is strongly concise in profinite groups.

######  Theorem 3.4.

Let $G$ be a profinite group in which the word $w_{o}$ takes less than
$2^{\aleph_{0}}$ values. Then the verbal subgroup $w_{o}(G)$ is finite.

###### Proof.

Choose a $2$-Sylow subgroup $P$ of $G$. In view of Lemma 3.3 observe that $P$
is soluble, say of derived length $l$. It follows that if $Q$ is any finite
homomorphic image of $G$, the insoluble length $\lambda(Q)$ is bounded in
terms of $l$ only. Applying now a combination of Lemma 2 and Lemma 3 of [21],
we obtain that the group $G$ has a normal series of finite length

$1=G_{0}\leq G_{1}\leq\cdots\leq G_{h}=G$ (3.1)

each of whose factors is either prosoluble or a Cartesian product of non-
abelian finite simple groups.

Lemma 3.2 shows that the non-prosoluble factors of the above series are
finite. Let $C$ be the intersection of the centralizers in $G$ of the non-
prosoluble factors, that is

$C=\\{g\in G;\ [G_{i+1},g]\leq G_{i}\text{ whenever }G_{i+1}/G_{i}\text{ is
not prosoluble}\\}.$

Since the non-prosoluble factors are finite, it follows that $C$ is open in
$G$. Note that all the sections of the series obtained by intersecting $C$
with the series 3.1 are prosoluble. Therefore we conclude that $C$ is
prosoluble and hence Lemma 3.3 tells us that $C^{\prime}$ is finite. Passing
to the quotient $G/C^{\prime}$ without loss of generality we can assume that
our group $G$ is virtually abelian. Now the result follows from a theorem of
Detomi that says that every word is strongly concise in the class of virtually
nilpotent profinite groups [3].

Alternatively, we can argue in a more direct way. Since $G$ is virtually
abelian, the verbal subgroup $w_{o}(G)$ is generated by its normal open
abelian subgroup $A$ and finitely many $w_{o}$-values, say
$g_{1},\dots,g_{s}$. Here, by Lemma 2.2, the centralizer $C_{G}(g_{i})$ is
open for every $i=1,\dots,s$. It follows that $B=A\cap
C_{G}(g_{1})\cap\dots\cap C_{G}(g_{s})$ is open in $w_{o}(G)$. Obviously,
$B\leq Z(w_{o}(G))$ so we conclude that $Z(w_{o}(G))$ is open in $w_{o}(G)$.
The Schur theorem tells us that the commutator subgroup $w_{o}(G)^{\prime}$ is
finite so we can pass to the quotient $G/w_{o}(G)^{\prime}$ and without loss
of generality assume that $w_{o}(G)$ is abelian. But then $G$ is metabelian
and the result is immediate from Lemma 3.3.

∎

## References

  * [1] Acciarri, Cristina and Shumyatsky, Pavel, On words that are concise in residually finite groups. J. Pure Appl. Algebra 218 (2014), 130–134.
  * [2] Azevedo, Joao and Shumyatsky, Pavel, On finiteness of some verbal subgroups in profinite groups. J. Algebra 574 (2021), 573–583.
  * [3] Detomi, Eloisa, A note on strong conciseness in virtually nilpotent profinite groups. _Arch. Math. (Basel)_ , 120(2) (2023), 115–121.
  * [4] Detomi, Eloisa and Klopsch, Benjamin and Shumyatsky, Pavel, Strong conciseness in profinite groups. J. Lond. Math. Soc. 102(3) (2020), 977–993.
  * [5] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Words of Engel type are concise in residually finite groups. Bulletin of Mathematical Sciences 9 (2019) 1950012 (19 pages).
  * [6] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Words of Engel type are concise in residually finite groups. Part II. Groups Geom. Dyn. 14 (2020), 991-1005.
  * [7] Detomi, Eloisa, and Morigi, Marta and Shumyatsky, Pavel, On bounded conciseness of Engel-like words in residually finite groups. J. Algebra 521 (2019), 1–15.
  * [8] Fernández-Alcober, Gustavo A. and Morigi, Marta, Outer commutator words are uniformly concise, J. Lond. Math. Soc. 82(3) (2010), 581–595.
  * [9] Fernández-Alcober, Gustavo A. and Pintonello, Matteo, Conciseness on normal subgroups and new concise words from lower central and derived words. Accepted in J. Iranian Math. Soc..
  * [10] Fernández-Alcober, Gustavo A. and Shumyatsky, Pavel, On bounded conciseness of words in residually finite groups. J. Algebra 500 (2018), 19–29.
  * [11] Guralnick, Robert and Shumyatsky, Pavel, On rational and concise word. J. Algebra 429 (2015), 213–217.
  * [12] de las Heras, Iker and Pintonello, Matteo and Shumyatsky, Pavel, Strong conciseness of coprime commutators in profinite groups. Accepted in J. Algebra
  * [13] Ivanov, Sergej, P. Hall’s conjecture on the finiteness of verbal subgroups. _Soviet Math. (Iz. VUZ)_ 33 (1989), 59–70.
  * [14] Jaikin-Zapirain, Andrei, On the verbal width of finitely generated pro-p groups. Rev. Mat. Iberoam. 168 (2008), 393–412.
  * [15] Khukhro, Evgeny and Shumyatsky, Pavel, Nonsoluble and non-p-soluble length of finite groups. Israel J. Math. 207(2) (2015), 507–525.
  * [16] Khukhro, Evgeny and Shumyatsky, Pavel, Strong conciseness of Engel words in profinite groups, Mathematische Nachrichten, 296(6) (2023), 2404–2416.
  * [17] Ol’shanskii, Alexander Y., Varieties in which all finite groups are abelian. Mat. Sb. (N.S.) 126(168)(1) (1985) 59–82.
  * [18] Ol’shanskii, Alexander Y., Geometry of Defining Relations in Groups. Mathematics and its applications 70 (Soviet Series), Kluwer Academic Publishers, Dordrecht, 1991.
  * [19] Segal, Dan, Words: notes on verbal width in groups. LMS Lecture Notes 361, Cambridge Univ. Press, Cambridge, 2009.
  * [20] Segal, Dan and Shalev, Aner, On groups with bounded conjugacy classes. Quart. J. Math. Oxford Ser 50(200) (1999), 505–516
  * [21] Wilson, John, On the structure of compact torsion groups, Monatsh. Math. 96(1) (1983), 57–66.