# Applications of the Schwarzschild-Finsler-Randers model

E. Kapsabelis<EMAIL_ADDRESS>Section of Astrophysics, Astronomy
and Mechanics, Department of Physics, National and Kapodistrian University of
Athens, Panepistimiopolis 15784, Athens, Greece  A. Triantafyllopoulos
<EMAIL_ADDRESS>Section of Astrophysics, Astronomy and Mechanics,
Department of Physics, National and Kapodistrian University of Athens,
Panepistimiopolis 15784, Athens, Greece  S. Basilakos
<EMAIL_ADDRESS>Academy of Athens, Research Center for Astronomy and
Applied Mathematics, Soranou Efessiou 4, 115 27 Athens, Greece  National
Observatory of Athens, Lofos Nymfon, 11852, Athens, Greece  P. C. Stavrinos
<EMAIL_ADDRESS>Department of Mathematics, National and Kapodistrian
University of Athens, Panepistimiopolis 15784, Athens, Greece

###### Abstract

In this article, we study further applications of the Schwarzschild-Finsler-
Randers (SFR) model which was introduced in a previous work [40]. In this
model, we investigate curvatures and the generalized Kretschmann invariant
which plays a crucial role for singularities. In addition, the derived path
equations are used for the gravitational redshift of the SFR-model and these
are compared with the GR model. Finally, we get some results for different
values of parameters of the generalized photonsphere of the SFR-model and we
find small deviations from the classical results of general relativity (GR)
which may be ought to the possible Lorentz violation effects.

###### pacs:

04.50.-h, 04.50.Kd

##  I Introduction

The last decade has seen a rapid increase of Finsler and Finsler-like
geometries and their applications to gravitation and cosmology with
appreciable results in the scientific community. We quote some relevant works
which have contributed in the development of applications of Finsler and
Finsler-like geometries to the gravitational field theory and cosmology [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49].

Finsler geometry is a dynamical metric geometry depending on position and
direction or dynamical coordinates on a tangent or fiber bundle of a
differentiable manifold. This type of geometry can also be connected to
Lorentz violation investigations of the standard model extension (SME) [50,
51] and in the context of local anisotropy [4, 8, 11, 19]. Moreover, Finsler-
like geometries breaking the local four dimensional Lorentz invariance can be
considered as a possible alternative direction for investigating physical
models with both local anisotropy and violation of local spacetime symmetries
[3].

A significant class of Finslerian spacetime is the Finsler-Randers (FR)
spacetime proposed by Randers [52]. An FR space has a metric function of the
form

$F(x,y)=(-a_{\mu\nu}(x)y^{\mu}y^{\nu})^{1/2}+u_{\alpha}y^{\alpha}$ (1)

where $u_{\alpha}$ is a covector with $||u_{\alpha}||\ll 1$,
$y^{\alpha}=\frac{dx^{\alpha}}{d\tau}$ and $a_{\mu\nu}(x)$ is a Riemannian
metric for which the Lorentzian signature $(-,+,+,+)$ has been assumed and the
indices $\mu,\nu,\alpha$ take the values $0,1,2,3$. The geodesics of this
space can be produced by (1) and the Euler-Lagrange equations. If $u_{\alpha}$
denotes a force field $f_{\alpha}$ and $y^{\alpha}$ is substituted with
$dx^{\alpha}$ then $f_{\alpha}dx^{\alpha}$ represents the spacetime effective
energy produced by the anisotropic force field $f_{\alpha}$, therefore
equation (1) is written as

$F(x,dx)=\left(-a_{\mu\nu}(x)dx^{\mu}dx^{\nu}\right)^{1/2}+f_{\alpha}dx^{\alpha}$
(2)

The integral $\int_{a}^{b}F(x,dx)$ represents the total work that some
particle needs to move along a path.

The length of a curve $c$ in the FR space is given by

$l(c)=\int^{1}_{0}F(x,\dot{x})d\tau$ (3)

where $\dot{x}=\frac{dx}{d\tau}$ and $\tau$ is affine parameter.

An FR cosmological model was introduced and studied in [2], [5]. In this case,
by considering the metric of the FRW cosmological model instead of
$a_{\mu\nu}(x)$ in (2) we get

$a_{\mu\nu}(x)=\mathrm{diag}\left[-1,\frac{a^{2}}{1-\kappa
r^{2}},a^{2}r^{2},a^{2}r^{2}\sin^{2}\theta\right]$ (4)

and we obtain a Finsler-Randers cosmology. From (2) we can notice that an FR
spacetime shows a motion of the FRW model with a produced work which comes
from the second term (one-form). This form of metric provides a dynamic
effective structure in spacetime. More investigations about this model can be
found in the following articles [6, 15, 30, 53, 54, 55, 56, 57, 58, 59, 60].

By using a Schwarzschild metric in (1), we obtain a Schwarzschild - Randers
spacetime [40].

$\displaystyle F(x,y)=$
$\displaystyle\left[-\left(1-\frac{R_{s}}{r}\right)(y^{t})^{2}+\frac{(y^{r})^{2}}{1-\frac{R_{s}}{r}}+r^{2}(y^{\theta})^{2}\right.+$
$\displaystyle+r^{2}\sin^{2}\theta\,(y^{\phi})^{2}\Bigg{]}^{1/2}+u_{\alpha}y^{\alpha}$
(5)

From (I), we can also see that the Schwarzschild-Randers metric has a
dynamical second term.

Finsler and Finsler-Randers spacetimes can give an effective description of
fermion particles with CPT-odd Lorentz violating terms in the SME framework
[19, 55, 61].

In this work, we elaborate some fundamental results of the SFR model and
compare them with the corresponding ones of GR. We prove that the
gravitational redshift predicted from our model remains invariant compared
with the one of GR. Nevertheless, in the case of photon sphere, we find
infinitesimal deviations from GR which may be ought to the small anisotropic
perturbations coming from Lorentz violation effects. In addition, in our
generalized metric space, we calculate the Kretschmann invariants of the model
and we find that the generalized second Kretschmann invariant $K_{V}$ provides
more information for singularities with additional degrees of freedom.

This article is organized as follows: In sec. II we give some basic elements
from the geometry of SFR. In sec. III we present the curvatures and the field
equations. In sec. IV, V, VI and VII we give some applications of the SFR
model including paths, energy, gravitational redshift and photonsphere.
Finally, in the last section VIII we summarize the results of our work.

##  II Basic structure of the model

In this section, we briefly present the underlying geometry of the SFR
gravitational model, as well as the field equations for the SFR metric. The
solution of these equations for this metric is presented at the end of the
section. An extended study of this model can be found in [40, 44].

The Lorentz tangent bundle $TM$ is a fibered 8-dimensional manifold with local
coordinates $\\{x^{\mu},y^{\alpha}\\}$ where the indices of the $x$ variables
are $\kappa,\lambda,\mu,\nu,\ldots=0,\ldots,3$ and the indices of the $y$
variables are $\alpha,\beta,\ldots,\theta=4,\ldots,7$. The tangent space at a
point of $TM$ is spanned by the so called adapted basis
$\\{E_{A}\\}=\,\\{\delta_{\mu},\dot{\partial}_{\alpha}\\}$ with

$\delta_{\mu}=\dfrac{\delta}{\delta x^{\mu}}=\dfrac{\partial}{\partial
x^{\mu}}-N^{\alpha}_{\mu}(x,y)\dfrac{\partial}{\partial y^{\alpha}}$ (6)

and

$\dot{\partial}_{\alpha}=\dfrac{\partial}{\partial y^{\alpha}}$ (7)

where $N^{\alpha}_{\mu}$ are the components of a nonlinear connection
$N=N^{\alpha}_{\mu}(x,y)\,\mathrm{d}x^{\mu}\otimes\dot{\partial}_{\alpha}$.

The nonlinear connection induces a split of the total space $TTM$ into a
horizontal distribution $T_{H}TM$ and a vertical distribution $T_{V}TM$. The
above-mentioned split is expressed with the Whitney sum:

$TTM=T_{H}TM\oplus T_{V}TM$ (8)

The anholonomy coefficients of the nonlinear connection are defined as

$\Omega^{\alpha}_{\nu\kappa}=\dfrac{\delta N^{\alpha}_{\nu}}{\delta
x^{\kappa}}-\dfrac{\delta N^{\alpha}_{\kappa}}{\delta x^{\nu}}$ (9)

A Sasaki-type metric $\mathcal{G}$ on $TM$ is:

$\mathcal{G}=g_{\mu\nu}(x,y)\,\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}+v_{\alpha\beta}(x,y)\,\delta
y^{\alpha}\otimes\delta y^{\beta}$ (10)

We define the metrics $g_{\mu\nu}$ and $v_{\alpha\beta}$ to be pseudo-
Finslerian.

A pseudo-Finslerian metric $f_{\alpha\beta}(x,y)$ is defined as one that has a
Lorentzian signature of $(-,+,+,+)$ and that also obeys the following form:

$\displaystyle
f_{\alpha\beta}(x,y)=\pm\frac{1}{2}\dfrac{\partial^{2}F^{2}}{\partial
y^{\alpha}\partial y^{\beta}}$ (11)

where the function $F$ satisfies the following conditions [12]:

  1. 1.

$F$ is continuous on $TM$ and smooth on $\widetilde{TM}\equiv
TM\setminus\\{0\\}$ i.e. the tangent bundle minus the null set $\\{(x,y)\in
TM|F(x,y)=0\\}$ .

  2. 2.

$F$ is positively homogeneous of first degree on its second argument:

$F(x^{\mu},ky^{\alpha})=kF(x^{\mu},y^{\alpha}),\qquad k>0$ (12)

  3. 3.

The form

$f_{\alpha\beta}(x,y)=\pm\dfrac{1}{2}\dfrac{\partial^{2}F^{2}}{\partial
y^{\alpha}\partial y^{\beta}}$ (13)

defines a non-degenerate matrix:

$\det\left[f_{\alpha\beta}\right]\neq 0$ (14)

where the plus-minus sign in (11) is chosen so that the metric has the correct
signature.

In this work, we will follow the model presented in [40]. The metric
$g_{\mu\nu}$ is the classic Schwarzschild one:

$\displaystyle g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$
$\displaystyle\quad=-fdt^{2}+\frac{dr^{2}}{f}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}$
(15)

with $f=1-\frac{R_{s}}{r}$ and $R_{s}=2GM$ the Schwarzschild radius (we assume
units where $c=1$).

Hereafter, we consider an $\alpha$-Randers type metric as the one in rel.(1)
which is distinguished from the $\beta$-Randers type metric that is
investigated in the SME [11, 19, 23, 27].

The metric $v_{\alpha\beta}$ is derived from a metric function $F_{v}$ of the
$\alpha$-Randers type:

$F_{v}=\sqrt{-g_{\alpha\beta}(x)y^{\alpha}y^{\beta}}+A_{\gamma}(x)y^{\gamma}$
(16)

where
$g_{\alpha\beta}=g_{\mu\nu}\tilde{\delta}^{\mu}_{\alpha}\tilde{\delta}^{\nu}_{\beta}$
is the Schwarzschild metric and $A_{\gamma}(x)$ is a covector which expresses
a deviation from general relativity, with $|A_{\gamma}(x)|\ll 1$. The
nonlinear connection will take the form:

$N^{\alpha}_{\mu}=\frac{1}{2}y^{\beta}g^{\alpha\gamma}\partial_{\mu}g_{\beta\gamma}$
(17)

The metric tensor $v_{\alpha\beta}$ of (16) is derived from (11) after
omitting higher order terms $O(A^{2})$:

$v_{\alpha\beta}(x,y)=g_{\alpha\beta}(x)+w_{\alpha\beta}(x,y)$ (18)

where

$\displaystyle w_{\alpha\beta}=$
$\displaystyle\,\frac{1}{\tilde{a}}(A_{\beta}g_{\alpha\gamma}y^{\gamma}+A_{\gamma}g_{\alpha\beta}y^{\gamma}+A_{\alpha}g_{\beta\gamma}y^{\gamma})$
$\displaystyle+\frac{1}{\tilde{a}^{3}}A_{\gamma}g_{\alpha\epsilon}g_{\beta\delta}y^{\gamma}y^{\delta}y^{\epsilon}$
(19)

with $\tilde{a}=\sqrt{-g_{\alpha\beta}y^{\alpha}y^{\beta}}$. The total metric
defined in the previous steps is called the Schwarzschild-Finsler-Randers
(SFR) metric.

In this work, we consider a distinguished connection ($d-$connection) ${D}$ on
$TM$. This is a linear connection with coefficients
$\\{\Gamma^{A}_{BC}\\}=\\{L^{\mu}_{\nu\kappa},L^{\alpha}_{\beta\kappa},C^{\mu}_{\nu\gamma},C^{\alpha}_{\beta\gamma}\\}$
which preserves by parallelism the horizontal and vertical distributions:

$\displaystyle{D_{\delta_{\kappa}}\delta_{\nu}=L^{\mu}_{\nu\kappa}(x,y)\delta_{\mu}}\quad$
$\displaystyle,\quad
D_{\dot{\partial}_{\gamma}}\delta_{\nu}=C^{\mu}_{\nu\gamma}(x,y)\delta_{\mu}$
(20)
$\displaystyle{D_{\delta_{\kappa}}\dot{\partial}_{\beta}=L^{\alpha}_{\beta\kappa}(x,y)\dot{\partial}_{\alpha}}\quad$
$\displaystyle,\quad
D_{\dot{\partial}_{\gamma}}\dot{\partial}_{\beta}=C^{\alpha}_{\beta\gamma}(x,y)\dot{\partial}_{\alpha}$
(21)

From these, the definitions for partial covariant differentiation follow as
usual, e.g. for $X\in TTM$ we have the definitions for covariant h-derivative

$X^{A}_{|\nu}\equiv D_{\nu}\,X^{A}\equiv\delta_{\nu}X^{A}+L^{A}_{B\nu}X^{B}$
(22)

and covariant v-derivative

$X^{A}|_{\beta}\equiv
D_{\beta}\,X^{A}\equiv\dot{\partial}_{\beta}X^{A}+C^{A}_{B\beta}X^{B}$ (23)

The $d-$connection is metric-compatible when the following conditions are met:

$D_{\kappa}\,g_{\mu\nu}=0,\quad D_{\kappa}\,v_{\alpha\beta}=0,\quad
D_{\gamma}\,g_{\mu\nu}=0,\quad D_{\gamma}\,v_{\alpha\beta}=0$ (24)

A $d-$connection can be uniquely defined given that the following conditions
are satisfied:

  * •

The $d-$connection is metric compatible

  * •

Coefficients
$L^{\mu}_{\nu\kappa},L^{\alpha}_{\beta\kappa},C^{\mu}_{\nu\gamma},C^{\alpha}_{\beta\gamma}$
depend solely on the quantities $g_{\mu\nu}$, $v_{\alpha\beta}$ and
$N^{\alpha}_{\mu}$

  * •

Coefficients $L^{\mu}_{\kappa\nu}$ and $C^{\alpha}_{\beta\gamma}$ are
symmetric on the lower indices, i.e.
$L^{\mu}_{[\kappa\nu]}=C^{\alpha}_{[\beta\gamma]}=0$

We use the symbol $\mathcal{D}$ instead of $D$ for a connection satisfying the
above conditions, and call it a canonical and distinguished $d-$connection.
The coefficients of canonical and distinguished $d-$connection are

$\displaystyle L^{\mu}_{\nu\kappa}$
$\displaystyle=\frac{1}{2}g^{\mu\rho}\left(\delta_{k}g_{\rho\nu}+\delta_{\nu}g_{\rho\kappa}-\delta_{\rho}g_{\nu\kappa}\right)$
(25) $\displaystyle L^{\alpha}_{\beta\kappa}$
$\displaystyle=\dot{\partial}_{\beta}N^{\alpha}_{\kappa}+\frac{1}{2}v^{\alpha\gamma}\left(\delta_{\kappa}v_{\beta\gamma}-v_{\delta\gamma}\,\dot{\partial}_{\beta}N^{\delta}_{\kappa}-v_{\beta\delta}\,\dot{\partial}_{\gamma}N^{\delta}_{\kappa}\right)$
(26) $\displaystyle C^{\mu}_{\nu\gamma}$
$\displaystyle=\frac{1}{2}g^{\mu\rho}\dot{\partial}_{\gamma}g_{\rho\nu}$ (27)
$\displaystyle C^{\alpha}_{\beta\gamma}$
$\displaystyle=\frac{1}{2}v^{\alpha\delta}\left(\dot{\partial}_{\gamma}v_{\delta\beta}+\dot{\partial}_{\beta}v_{\delta\gamma}-\dot{\partial}_{\delta}v_{\beta\gamma}\right)$
(28)

Curvatures and torsions on $TM$ can be defined by the multilinear maps:

$\mathcal{R}(X,Y)Z=[\mathcal{D}_{X},\mathcal{D}_{Y}]Z-\mathcal{D}_{[X,Y]}Z$
(29)

and

$\mathcal{T}(X,Y)=\mathcal{D}_{X}Y-\mathcal{D}_{Y}X-[X,Y]$ (30)

where $X,Y,Z\in TTM$. We use the following definitions for the curvature
components [1, 3]:

$\displaystyle\mathcal{R}(\delta_{\lambda},\delta_{\kappa})\delta_{\nu}=R^{\mu}_{\nu\kappa\lambda}\delta_{\mu}$
(31)
$\displaystyle\mathcal{R}(\delta_{\lambda},\delta_{\kappa})\dot{\partial}_{\beta}=R^{\alpha}_{\beta\kappa\lambda}\dot{\partial}_{\alpha}$
(32)
$\displaystyle\mathcal{R}(\dot{\partial}_{\gamma},\delta_{\kappa})\delta_{\nu}=P^{\mu}_{\nu\kappa\gamma}\delta_{\mu}$
(33)
$\displaystyle\mathcal{R}(\dot{\partial}_{\gamma},\delta_{\kappa})\dot{\partial}_{\beta}=P^{\alpha}_{\beta\kappa\gamma}\dot{\partial}_{\alpha}$
(34)
$\displaystyle\mathcal{R}(\dot{\partial}_{\delta},\dot{\partial}_{\gamma})\delta_{\nu}=S^{\mu}_{\nu\gamma\delta}\delta_{\mu}$
(35)
$\displaystyle\mathcal{R}(\dot{\partial}_{\delta},\dot{\partial}_{\gamma})\dot{\partial}_{\beta}=S^{\alpha}_{\beta\gamma\delta}\dot{\partial}_{\alpha}$
(36)

In addition, we use the following definitions for the torsion components:

$\displaystyle\mathcal{T}(\delta_{\kappa},\delta_{\nu})=$
$\displaystyle\mathcal{T}^{\mu}_{\nu\kappa}\delta_{\mu}+\mathcal{T}^{\alpha}_{\nu\kappa}\dot{\partial}_{\alpha}$
(37) $\displaystyle\mathcal{T}(\dot{\partial}_{\beta},\delta_{\nu})=$
$\displaystyle\mathcal{T}^{\mu}_{\nu\beta}\delta_{\mu}+\mathcal{T}^{\alpha}_{\nu\beta}\dot{\partial}_{\alpha}$
(38)
$\displaystyle\mathcal{T}(\dot{\partial}_{\gamma},\dot{\partial}_{\beta})=$
$\displaystyle\mathcal{T}^{\mu}_{\beta\gamma}\delta_{\mu}+\mathcal{T}^{\alpha}_{\beta\gamma}\dot{\partial}_{\alpha}$
(39)

The h-curvature tensor of the $d-$connection in the adapted basis and the
corresponding h-Ricci tensor have, respectively, the components given from
(31):

$\displaystyle
R^{\mu}_{\nu\kappa\lambda}=\delta_{\lambda}L^{\mu}_{\nu\kappa}-\delta_{\kappa}L^{\mu}_{\nu\lambda}+L^{\rho}_{\nu\kappa}L^{\mu}_{\rho\lambda}-L^{\rho}_{\nu\lambda}L^{\mu}_{\rho\kappa}+C^{\mu}_{\nu\alpha}\Omega^{\alpha}_{\kappa\lambda}$
(40) $\displaystyle
R_{\mu\nu}=R^{\kappa}_{\mu\nu\kappa}=\delta_{\kappa}L^{\kappa}_{\mu\nu}-\delta_{\nu}L^{\kappa}_{\mu\kappa}+L^{\rho}_{\mu\nu}L^{\kappa}_{\rho\kappa}-L^{\rho}_{\mu\kappa}L^{\kappa}_{\rho\nu}$
$\displaystyle\quad\quad\quad\quad\quad\quad+C^{\kappa}_{\mu\alpha}\Omega^{\alpha}_{\nu\kappa}$
(41)

The v-curvature tensor of the $d-$connection in the adapted basis and the
corresponding v-Ricci tensor have, respectively, the components (36):

$\displaystyle S^{\alpha}_{\beta\gamma\delta}$
$\displaystyle=\dot{\partial}_{\delta}C^{\alpha}_{\beta\gamma}-\dot{\partial}_{\gamma}C^{\alpha}_{\beta\delta}+C^{\epsilon}_{\beta\gamma}C^{\alpha}_{\epsilon\delta}-C^{\epsilon}_{\beta\delta}C^{\alpha}_{\epsilon\gamma}$
(42) $\displaystyle S_{\alpha\beta}$
$\displaystyle=S^{\gamma}_{\alpha\beta\gamma}=\dot{\partial}_{\gamma}C^{\gamma}_{\alpha\beta}-\dot{\partial}_{\beta}C^{\gamma}_{\alpha\gamma}+C^{\epsilon}_{\alpha\beta}C^{\gamma}_{\epsilon\gamma}-C^{\epsilon}_{\alpha\gamma}C^{\gamma}_{\epsilon\beta}$
(43)

The generalized Ricci scalar curvature in the adapted basis is defined as

$\mathcal{R}=g^{\mu\nu}R_{\mu\nu}+v^{\alpha\beta}S_{\alpha\beta}=R+S$ (44)

where

$\displaystyle R=g^{\mu\nu}R_{\mu\nu}\quad,\quad
S=v^{\alpha\beta}S_{\alpha\beta}$ (45)

A Hilbert-like action on $TM$ can be defined as

$K=\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{R}+2\kappa\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}$
(46)

for some closed subspace $\mathcal{N}\subset TM$, where $|\mathcal{G}|$ is the
absolute value of the metric determinant, $\mathcal{L}_{M}$ is the Lagrangian
of the matter fields, $\kappa$ is a constant and

$d^{8}\mathcal{U}=\mathrm{d}x^{0}\wedge\ldots\wedge\mathrm{d}x^{3}\wedge\mathrm{d}y^{4}\wedge\ldots\wedge\mathrm{d}y^{7}$
(47)

Variation with respect to $g_{\mu\nu}$, $v_{\alpha\beta}$ and
$N^{\alpha}_{\kappa}$ leads to the following field equations [44] (see
Appendix A for more details):

$\displaystyle\overline{R}_{\mu\nu}-\frac{1}{2}({R}+{S})\,{g_{\mu\nu}}$
$\displaystyle\quad+\left(\delta^{(\lambda}_{\nu}\delta^{\kappa)}_{\mu}-g^{\kappa\lambda}g_{\mu\nu}\right)\left(\mathcal{D}_{\kappa}\mathcal{T}^{\beta}_{\lambda\beta}-\mathcal{T}^{\gamma}_{\kappa\gamma}\mathcal{T}^{\beta}_{\lambda\beta}\right)=\kappa{T_{\mu\nu}}$
(48) $\displaystyle S_{\alpha\beta}-\frac{1}{2}({R}+{S})\,{v_{\alpha\beta}}$
$\displaystyle\quad+\left(v^{\gamma\delta}v_{\alpha\beta}-\delta^{(\gamma}_{\alpha}\delta^{\delta)}_{\beta}\right)\left(\mathcal{D}_{\gamma}C^{\mu}_{\mu\delta}-C^{\nu}_{\nu\gamma}C^{\mu}_{\mu\delta}\right)=\kappa{Y_{\alpha\beta}}$
(49) $\displaystyle
g^{\mu[\kappa}\dot{\partial}_{\alpha}L^{\nu]}_{\mu\nu}+2\mathcal{T}^{\beta}_{\mu\beta}g^{\mu[\kappa}C^{\lambda]}_{\lambda\alpha}=\frac{\kappa}{2}\mathcal{Z}^{\kappa}_{\alpha}$
(50)

with

$\displaystyle T_{\mu\nu}$
$\displaystyle\equiv-\frac{2}{\sqrt{|\mathcal{G}|}}\frac{\Delta\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)}{\Delta
g^{\mu\nu}}=-\frac{2}{\sqrt{-g}}\frac{\Delta\left(\sqrt{-g}\,\mathcal{L}_{M}\right)}{\Delta
g^{\mu\nu}}$ (51) $\displaystyle Y_{\alpha\beta}$
$\displaystyle\equiv-\frac{2}{\sqrt{|\mathcal{G}|}}\frac{\Delta\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)}{\Delta
v^{\alpha\beta}}=-\frac{2}{\sqrt{-v}}\frac{\Delta\left(\sqrt{-v}\,\mathcal{L}_{M}\right)}{\Delta
v^{\alpha\beta}}$ (52) $\displaystyle\mathcal{Z}^{\kappa}_{\alpha}$
$\displaystyle\equiv-\frac{2}{\sqrt{|\mathcal{G}|}}\frac{\Delta\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)}{\Delta
N^{\alpha}_{\kappa}}=-2\frac{\Delta\mathcal{L}_{M}}{\Delta
N^{\alpha}_{\kappa}}$ (53)

where $\mathcal{L}_{M}$ is the Lagrangian of the matter fields,
$\delta^{\mu}_{\nu}$ and $\delta^{\alpha}_{\beta}$ are the Kronecker symbols,
$|\mathcal{G}|$ is the absolute value of the determinant of the total metric
(10), and

$\mathcal{T}_{\nu\beta}^{\alpha}=\dot{\partial}_{\beta}N_{\nu}^{\alpha}-L_{\beta\nu}^{\alpha}$
(54)

are torsion components, where $L_{\beta\nu}^{\alpha}$ is defined in (26). From
the form of (10) it follows that $\sqrt{|\mathcal{G}|}=\sqrt{-g}\sqrt{-v}$,
with $g,v$ the determinants of the metrics $g_{\mu\nu},v_{\alpha\beta}$
respectively.

Solving the above equations to first order in $A_{\gamma}(x)$ in vacuum
($T_{\mu\nu}=Y_{\alpha\beta}=\mathcal{Z}^{\kappa}_{\alpha}=0$), we get [40]:

$A_{\gamma}(x)=\left[\tilde{A}_{0}\left|1-\frac{R_{S}}{r}\right|^{1/2},0,0,0\right]$
(55)

with $\tilde{A}_{0}$ a constant.

##  III Curvatures and generalized Kretschmann invariants

It is useful to calculate invariants of the metrics $g_{\mu\nu}$ and
$v_{\alpha\beta}$ so that we can get a better understanding for the behaviour
of the solution in specific points of $TM$. Specifically, any point in $TM$
where these invariants diverge can be considered singular, namely, a point
where our geometrical model breaks down.

We consider invariants constructed from contractions of the curvature tensor
on $TM$ and the metric. The nonvanishing curvature components of the
distinguished canonical connection are given by relations (31)-(36):

$\displaystyle R^{\mu}_{\nu\kappa\lambda}$
$\displaystyle=\delta_{\lambda}L^{\mu}_{\nu\kappa}-\delta_{\kappa}L^{\mu}_{\nu\lambda}+L^{\rho}_{\nu\kappa}L^{\mu}_{\rho\lambda}-L^{\rho}_{\nu\lambda}L^{\mu}_{\rho\kappa}+C^{\mu}_{\nu\alpha}\Omega^{\alpha}_{\kappa\lambda}$
(56) $\displaystyle R^{\alpha}_{\beta\kappa\lambda}$
$\displaystyle=\delta_{\lambda}L^{\alpha}_{\beta\kappa}-\delta_{\kappa}L^{\alpha}_{\beta\lambda}+L^{\gamma}_{\beta\kappa}L^{\alpha}_{\gamma\lambda}-L^{\gamma}_{\beta\lambda}L^{\alpha}_{\gamma\kappa}+C^{\alpha}_{\beta\gamma}\Omega^{\gamma}_{\kappa\lambda}$
(57) $\displaystyle P^{\mu}_{\nu\kappa\gamma}$
$\displaystyle=\dot{\partial}_{\gamma}L^{\mu}_{\nu\kappa}-\mathcal{D}_{\kappa}C^{\mu}_{\nu\gamma}+C^{\mu}_{\nu\beta}\mathcal{T}^{\beta}_{\kappa\gamma}$
(58) $\displaystyle P^{\alpha}_{\beta\kappa\gamma}$
$\displaystyle=\dot{\partial}_{\gamma}L^{\alpha}_{\beta\kappa}-\mathcal{D}_{\kappa}C^{\alpha}_{\beta\gamma}+C^{\alpha}_{\beta\delta}\mathcal{T}^{\delta}_{\kappa\gamma}$
(59) $\displaystyle S^{\mu}_{\nu\gamma\delta}$
$\displaystyle=\dot{\partial}_{\delta}C^{\mu}_{\nu\gamma}-\dot{\partial}_{\gamma}C^{\mu}_{\nu\delta}+C^{\kappa}_{\nu\gamma}C^{\mu}_{\kappa\delta}-C^{\kappa}_{\nu\delta}C^{\mu}_{\kappa\gamma}$
(60) $\displaystyle S^{\alpha}_{\beta\gamma\delta}$
$\displaystyle=\dot{\partial}_{\delta}C^{\alpha}_{\beta\gamma}-\dot{\partial}_{\gamma}C^{\alpha}_{\beta\delta}+C^{\epsilon}_{\beta\gamma}C^{\alpha}_{\epsilon\delta}-C^{\epsilon}_{\beta\delta}C^{\alpha}_{\epsilon\gamma}$
(61)

where $\mathcal{D}_{\kappa}$ is the covariant derivative with respect to the
connection defined in (25) - (28). An explicit calculation for the SFR metric
yields

$R^{\alpha}_{\beta\kappa\lambda}=P^{\mu}_{\nu\kappa\gamma}=P^{\alpha}_{\beta\kappa\gamma}=S^{\mu}_{\nu\gamma\delta}=0$
(62)

Consequently, we cannot construct any non-vanishing invariant of the metrics
from these components so they are not useful for finding singular points.
Additionally, we get $g^{\mu\nu}R_{\mu\nu}=0$ so this scalar curvature gives
no information about singular points either.

Next, we calculate the scalar curvature $S=v^{\alpha\beta}S_{\alpha\beta}$ to
the lowest non-vanishing order and we find:

$S=\frac{5\tilde{A}_{0}^{2}r\left\\{(y^{r})^{2}+r(r-R_{S})\left[(y^{\theta})^{2}+\sin^{2}\theta(y^{\phi})^{2}\right]\right\\}}{2\tilde{a}^{4}\left(r-R_{S}\right)}$
(63)

with $\tilde{a}=\sqrt{-g_{\alpha\beta}y^{\alpha}y^{\beta}}$ and we have set
$y^{4}\equiv y^{t},y^{5}\equiv y^{r},y^{6}\equiv y^{\theta},y^{7}\equiv
y^{\phi}$. From (63), we see that the anisotropic scalar curvature $S$ has a
geometrical meaning because of its dependence on the coordinates.

A straightforward calculation results in the following cases:

  1. 1.

$\tilde{a}\neq 0$, $r=R_{S}$ and $y^{r}\neq 0$: In this case, we get $S=0$

  2. 2.

$\tilde{a}\neq 0$, $r=R_{S}$ and $y^{r}=0$: In this case, the fiber scalar
curvature takes the value

$S=\frac{5\tilde{A}_{0}^{2}}{2R_{S}^{2}\left[(y^{\theta})^{2}+\sin^{2}\theta(y^{\phi})^{2}\right]}$
(64)

  3. 3.

$\tilde{a}=0$: In this case, the fiber scalar curvature diverges:
$S\rightarrow\infty$

The third case is the most interesting one, where it can be seen that
$\tilde{a}=0$ represents a set of singular points for the metric
$v_{\alpha\beta}$. In the next paragraphs, we will identify $y^{\alpha}$ with
the 4-velocity of a free particle, in which case the condition $\tilde{a}=0$
will denote a null path with respect to the metric $g_{\mu\nu}(x)$. Taking
this argument into account, we reach the conclusion that such paths can not
describe physical trajectories.

Finally, we calculate the nontrivial Kretschmann-like invariants of the
metrics $g_{\mu\nu}$ and $v_{\alpha\beta}$ to the lowest non-vanishing order:

$\displaystyle K_{H}$ $\displaystyle\equiv
R_{\kappa\lambda\mu\nu}R^{\kappa\lambda\mu\nu}=\frac{12R_{S}^{2}}{r^{6}}$ (65)
$\displaystyle K_{V}$ $\displaystyle\equiv
S_{\alpha\beta\gamma\delta}S^{\alpha\beta\gamma\delta}=\left(\frac{3S}{5}\right)^{2}$
(66)

The invariant in Eq.(65) coincides with the Kretschmann invariant of the
classic Schwarzschild solution [62] and it reveals a singularity of the metric
$g_{\mu\nu}$ at the point $r=0$. The second Kretschmann-like invariant
contains the same information as the scalar curvature $S$, as we can see from
Eqs.(63) and (66), so the same conclusions apply for it.

We notice from (65) and (66) that the total Kretschmann invariant
$K=K_{H}+K_{V}$ is equal to the classic Schwarzschild one plus a small
correction which comes from the additional geometrical inner structure of the
SFR gravitational model. Specifically, the scalar curvature of the vertical
space (the space of $y-$variables) is related to a non-trivial vertical-space
Kretschmann invariant, as one can see from (66), so it induces a deviation
from classical general relativity.

##  IV Paths

In this section, we study the paths of a particle in the SFR model. We
consider the Lagrangian of the form [44]:

$L(x,\dot{x},y)=\left(-ag_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}-b\tilde{\delta}^{\alpha}_{\mu}v_{\alpha\beta}\dot{x}^{\mu}y^{\beta}-cv_{\alpha\beta}y^{\alpha}y^{\beta}\right)^{1/2}$
(67)

with $a,b,c$ constants. Variation of the action with respect to $y^{\alpha}$
gives the relation:

$y^{\alpha}=\dot{x}^{\alpha}$ (68)

Furthermore, if we variate the action with respect to $x^{\mu}$ and substitute
(68), we get the path equations:

$\displaystyle\ddot{x}^{\mu}+\gamma^{\mu}_{\kappa\lambda}\dot{x}^{\kappa}\dot{x}^{\lambda}$
$\displaystyle=-\frac{z}{1+z}\Big{\\{}\tilde{a}g^{\mu\nu}\left(\partial_{\nu}A_{\kappa}-\partial_{\kappa}A_{\nu}\right)\dot{x}^{\kappa}$
$\displaystyle+\frac{1}{\tilde{a}}\Big{[}A^{\nu}\left(\partial_{\kappa}g_{\nu\lambda}-\frac{1}{2}\partial_{\nu}g_{\kappa\lambda}\right)+\partial_{\kappa}A_{\lambda}\Big{]}\dot{x}^{\mu}\dot{x}^{\kappa}\dot{x}^{\lambda}$
$\displaystyle+\frac{1}{\tilde{a}}\Big{(}\frac{1}{4}g^{\mu\nu}\\!A_{\kappa}\partial_{\nu}g_{\sigma\lambda}+g^{\mu\nu}\\!A_{\kappa}\partial_{\lambda}g_{\sigma\nu}+A^{\mu}\partial_{\kappa}g_{\lambda\sigma}\Big{)}\dot{x}^{\sigma}\dot{x}^{\kappa}\dot{x}^{\lambda}$
$\displaystyle+\frac{1}{2\tilde{a}^{3}}A_{\lambda}\partial_{\kappa}g_{\sigma\tau}\dot{x}^{\sigma}\dot{x}^{\tau}\dot{x}^{\mu}\dot{x}^{\kappa}\dot{x}^{\lambda}\Big{\\}}$
(69)

where $z=-b^{2}/4ac$ is a constant and a dot denotes differentiation with
respect to the generalized proper time $\tau$, with the definition

$d\tau=\left[-ag_{\mu\nu}dx^{\mu}dx^{\nu}-(b+c)v_{\alpha\beta}dx^{\alpha}dx^{\beta}\right]^{1/2}$
(70)

which is derived from the Lagrangian (67) if we substitute
$y^{\alpha}=dx^{\alpha}$. The form (IV) generalizes the geodesics equations of
general relativity in the SFR model.

In order to solve the equations (IV) we use the NDSolve command of Mathematica
to obtain a numerical timelike solution. By assuming different initial values
we get two different solutions which are described by a closed path and an
open path respectively and we compare our results with the geodesics of GR for
the same initial values. In our approach, we consider the energy
$E=\left(1-\frac{R_{s}}{r}\right)\dfrac{dt}{d\tau}$ and the angular momentum
$L=r^{2}\dfrac{d\phi}{d\tau}$ .

Fig. 1: This is an $r,t$ graph of the timelike paths that we find using our
theoretical SFR (red curve) model in comparison to the geodesics of GR (blue
line) for $E=0.98$, $L=1$, $r_{0}=3$ and $(a,b,c)=(1,1,1)$. Fig. 2: This is a
polar graph of the timelike paths that we find using our theoretical SFR (red
curve) model in comparison to the geodesics of GR (blue line) for $E=0.98$,
$L=1$, $r_{0}=3$ and $(a,b,c)=(1,1,1)$.

We notice from the two graphs (Fig.1 and Fig.2) that the paths in the SFR
model and GR are very similar. However, from the r-t graph (Fig.1) we can see
that the maximum radial distance in SFR is lower and the required time to
reach the Schwarzschild radius is also less compared to GR. From the second
graph (Fig.2) we can see that the two ellipses are similar but the red ellipse
(SFR model) is smaller and it reaches the event horizon faster than the blue
ellipse (GR). We remark that in the path equations (IV) the right hand side is
non-zero and this term acts as a small extra force that influences the paths
in the gravitational field. This correction increases or decreases the effects
of gravity depending on the sign of the term.

Fig. 3: This is an $r,t$ graph of the timelike paths that we find using our
theoretical SFR (red curve) model in comparison to the geodesics of GR (blue
line) for $E=0.98$, $L=1$, $r_{0}=3$ and $(a,b,c)=(1,10,10)$. Fig. 4: This is
a polar graph of the timelike paths that we find using our theoretical SFR
(red curve) model in comparison to the geodesics of GR (blue line) for
$E=0.98$, $L=1$, $r_{0}=3$ and $(a,b,c)=(1,10,10)$.

In the two figures (Fig.3 and Fig.4) we have taken $a=1$, $b=10$ and $c=10$ in
(67). In this case, we can see that the red line (SFR model) takes higher
values than the blue line (GR) and it requires more time to reach the
Schwarzschild radius. In our case, the parameters $(\tilde{A}_{0},a,b,c)$
control the deviation of the SFR model from General Relativity. In particular,
the values of $(a,b,c)$ can give higher or lower results compared to GR.

Fig. 5: This is an r-t graph of the timelike paths in the SFR model in
comparison to the geodesics in GR for $E=1.2$, $L=4$, $r_{0}=5$ and
$(a,b,c)=(1,10,10)$.

The last two graphs (Fig.5 and Fig.6) represent open paths with $a=1$, $b=10$
and $c=10$. In this case, we can see that the SFR model deviates from GR when
we start to move away from the event horizon and the two paths (red and blue)
separate. For a small interval, the paths of the SFR model approximate the
geodesics of GR. As the radial distance increases, the paths of our model
deviate from GR.

Fig. 6: This is a polar graph of the timelike paths in the SFR model in
comparison to the geodesics in GR for $E=1.2$, $L=4$, $r_{0}=5$ and
$(a,b,c)=(1,10,10)$.

##  V Energy

In this section, we give the form of the energy and momentum of a particle in
an SFR spacetime.

We assume a four-velocity vector
$u^{\alpha}=(u^{t},u^{r},u^{\theta},u^{\phi})$, with

$u^{\alpha}\equiv\dfrac{dx^{\alpha}}{d\tau}$ (71)

and we require that its norm equals $-1$, so we have [45]:

$||u||=u^{\alpha}u_{\alpha}=u^{\alpha}u^{\beta}v_{\alpha\beta}=-1$ (72)

By use of (18) we find111The condition (73) along with relation (70) give
$a=b+c=1$ in this case.:

$g_{\alpha\beta}u^{\alpha}u^{\alpha}+w_{\alpha\beta}u^{\alpha}u^{\beta}=-1$
(73)

and by (II) we get:

$\displaystyle
g_{\alpha\beta}u^{\alpha}u^{\beta}+\frac{1}{\tilde{a}}\Big{[}(A_{\beta}u^{\beta})(g_{\alpha\gamma}u^{\alpha}u^{\gamma})$
$\displaystyle+(A_{\gamma}u^{\gamma})(g_{\alpha\beta}u^{\alpha}u^{\beta})+(A_{\alpha}u^{\alpha}(g_{\beta\gamma}u^{\beta}u^{\gamma})\Big{]}$
$\displaystyle+\frac{1}{\tilde{a}^{3}}(A_{\gamma}u^{\gamma})(g_{\alpha\epsilon}u^{\alpha}u^{\epsilon})(g_{\beta\delta}u^{\beta}u^{\delta})=-1$
(74)

where we have set $y^{\alpha}=u^{\alpha}$ and
$\tilde{a}=\sqrt{-g_{\alpha\beta}u^{\alpha}u^{\beta}}$.

After some calculations, we have:

$\tilde{a}^{2}+2A_{\gamma}u^{\gamma}\tilde{a}-1=0$ (75)

By solving (75) we can find $\tilde{a}$

$\tilde{a}=-\tilde{A}_{0}f^{1/2}u^{t}+\sqrt{1+\tilde{A}_{0}^{2}f(u^{t})^{2}}$
(76)

where we have used (55) for $A_{\gamma}$ with $f=1-\frac{R_{S}}{r}$.
If we use a Taylor expansion for the second term and omit higher order terms
$O(\tilde{A_{0}}^{2})$ we get:

$\tilde{a}=1-\tilde{A}_{0}f^{1/2}u^{t}$ (77)

Equation (77) is the condition so that the norm of the four-velocity equals
$-1$.

If we assume that the particle is at rest, the four-velocity becomes
$u^{\alpha}=(u^{t},0,0,0)$ and if we substitute this in (77) we find:

$u^{t}_{SFR}=(1-\tilde{A}_{0})f^{-1/2}$ (78)

We see from (78) that if $\tilde{A}_{0}\rightarrow 0$ we find the result from
GR :

$u^{t}_{GR}=f^{-1/2}$ (79)

Consequently, by using (78) and (79) we can write:

$u^{t}_{SFR}=(1-\tilde{A}_{0})u^{t}_{GR}$ (80)

From rel.(80) we see that if $\tilde{A}_{0}$ has a positive value then
$u^{t}_{SFR}<u^{t}_{GR}$ and if $\tilde{A}_{0}$ has a negative value then
$u^{t}_{SFR}>u^{t}_{GR}$.
We can find the momentum and energy of the particle:

$p^{\alpha}=mu^{\alpha}=(mu^{t},0,0,0)$ (81)

where m is the mass of the particle.
From rel.(78) we get for $p^{t}_{SFR}$ and $E_{SFR}$:

$E_{SFR}=p^{t}_{SFR}=m(1-\tilde{A}_{0})f^{-1/2}$ (82)

##  VI Gravitational Redshift

If we take $r,\theta,\phi=const$ in the definition of proper time (70), we
get:

$d\tau=\left[-ag_{00}dt^{2}-(b+c)v_{00}dt^{2}\right]^{1/2}$ (83)

By using eq.(18), we get:

$d\tau=[-g_{00}-\kappa w_{00}]^{1/2}dt^{\prime}$ (84)

where we have set $dt^{\prime}=\sqrt{a+b+c}dt$ and $\kappa=\frac{b+c}{a+b+c}$.
From the definition of the metric perturbation $w_{\alpha\beta}$ in (II) for
$\alpha=0$ and $\beta=0$ we get:

$\displaystyle w_{00}=$
$\displaystyle\,\frac{1}{\tilde{a}}\left(A_{0}g_{00}\dot{x}^{0}+A_{0}g_{00}\dot{x}^{0}+A_{0}g_{00}\dot{x}^{0}\right)$
$\displaystyle+\frac{1}{\tilde{a}^{3}}A_{0}g_{00}g_{00}\dot{x}^{0}\dot{x}^{0}\dot{x}^{0}$
$\displaystyle\Rightarrow w_{00}=$ $\displaystyle\,-2\tilde{A}_{0}f$ (85)

where
$\tilde{a}=\sqrt{-g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}}=\dot{t}\sqrt{-g_{00}}=\dot{t}\sqrt{f}$
because we have taken $r,\theta,\phi=$const.

If we return to (84) we find:

$\displaystyle d\tau$ $\displaystyle=[-g_{00}-\kappa w_{00}]^{1/2}dt^{\prime}$
$\displaystyle\Rightarrow d\tau$
$\displaystyle=[f-\kappa(-2\tilde{A}_{0}f)]^{1/2}dt^{\prime}$
$\displaystyle\Rightarrow d\tau$
$\displaystyle=(1+\epsilon)^{1/2}\sqrt{-g_{00}}\,dt^{\prime}$ (86)

where we have set $\epsilon=2\kappa\tilde{A}_{0}$ with $\epsilon\ll 1$,
$g_{00}=-f$ and $f=1-\frac{R_{s}}{r}$. We note that in GR the calculation for
the redshift leads to : $d\tau_{GR}=\sqrt{-g_{00}}\,dt$.

Now, if we consider two clocks at two different points of spacetime $r_{1}$
and $r_{2}$, we will have:

$d\tau_{1}=(1+\epsilon)^{1/2}\sqrt{-g_{00}(1)}\,dt^{\prime}$ (87)

and

$d\tau_{2}=(1+\epsilon)^{1/2}\sqrt{-g_{00}(2)}\,dt^{\prime}$ (88)

and thus for the frequencies $\nu_{1}$ and $\nu_{2}$ we find:

$\displaystyle\nu_{2}$
$\displaystyle=\nu_{1}\left(\frac{g_{00}(1)}{g_{00}(2)}\right)^{1/2}=\nu_{1}\left(\frac{1-\frac{R_{s}}{r_{1}}}{1-\frac{R_{s}}{r_{2}}}\right)^{1/2}$
$\displaystyle\Rightarrow\frac{\nu_{2}}{\nu_{1}}$ $\displaystyle\approx
1-GM\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right)$ (89)

where we have used the Taylor expansion $(1+x)^{1/2}\approx 1+\frac{1}{2}x$.
From (89) we find :

$\left(\frac{\Delta\nu}{\nu_{1}}\right)_{SFR}=\Delta U$ (90)

where $\Delta\nu=\nu_{2}-\nu_{1}$ with $\nu_{2},\nu_{1}$ the emitter and
receiver frequencies and $\Delta U=GM(\frac{1}{r_{2}}-\frac{1}{r_{1}})$ is the
change of potential.

We recall that in general relativity (GR) the gravitational redshift is given
by:

$\left(\frac{\Delta\nu}{\nu_{1}}\right)_{GR}=\Delta U$ (91)

We remark that, in the scenario under consideration, the gravitational
redshift predicted by the SFR gravitational model is the same as the one
predicted in the classic Schwarzschild spacetime of GR.

##  VII Photonsphere

In order to calculate the radius of the photonsphere we will use eq.(70):

$d\tau=\left[-ag_{\mu\nu}dx^{\mu}dx^{\nu}-(b+c)v_{\alpha\beta}dx^{\alpha}dx^{\beta}\right]^{1/2}$
(92)

From (18) we get:

$d\tau^{\prime}=\left(-g_{\mu\nu}dx^{\mu}dx^{\nu}-\kappa
w_{\alpha\beta}dx^{\alpha}dx^{\beta}\right)^{1/2}$ (93)

where $\kappa=\frac{b+c}{a+b+c}$ and
$d\tau^{\prime}=\frac{d\tau}{\sqrt{a+b+c}}$.
To calculate the radius of the photonsphere, we take $r=$ const. ,
$\theta=\frac{\pi}{2}$ and $d\tau^{\prime}=0$ because we want to find the
photon orbits.
Under these conditions, rel.(93) yields:

$(g_{00}+\kappa w_{00})dt^{2}+(g_{33}+\kappa w_{33})d\phi^{2}=0$ (94)

From the above relation, we find:

$\left(\frac{d\phi}{dt}\right)^{2}=-\frac{g_{00}+\kappa w_{00}}{g_{33}+\kappa
w_{33}}$ (95)

To calculate $w_{00}$ and $w_{33}$, we use (II).

$\displaystyle w_{\alpha\beta}=$
$\displaystyle\,\frac{1}{\tilde{a}}(A_{\beta}g_{\alpha\gamma}y^{\gamma}+A_{\gamma}g_{\alpha\beta}y^{\gamma}+A_{\alpha}g_{\beta\gamma}y^{\gamma})$
$\displaystyle+\frac{1}{\tilde{a}^{3}}A_{\gamma}g_{\alpha\epsilon}g_{\beta\delta}y^{\gamma}y^{\delta}y^{\epsilon}$
(96)

where $\tilde{a}=\sqrt{-g_{\alpha\beta}y^{\alpha}y^{\beta}}$ and for
$A_{\gamma}$ we use (55).
We calculate $\tilde{a}$ :

$\displaystyle\tilde{a}=\sqrt{-g_{\alpha\beta}y^{\alpha}y^{\beta}}$
$\displaystyle=\sqrt{-g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}}=\sqrt{f\dot{t}^{2}-r^{2}\dot{\phi}^{2}}\Rightarrow$
$\displaystyle\tilde{a}=$ $\displaystyle\dot{t}\sqrt{f-r^{2}\phi^{\prime
2}}=\dot{t}\tilde{p}$ (97)

where we used the Leibniz chain rule
$\dot{\phi}=\frac{d\phi}{dt}\frac{dt}{d\tau}=\phi^{\prime}(t)\dot{t}$ and we
set $\tilde{p}=\sqrt{f-r^{2}\phi^{\prime 2}}$.
After some calculations, we find:

$\displaystyle w_{00}$
$\displaystyle=\frac{\tilde{A}_{0}}{\tilde{p}^{3}}f^{3/2}\big{(}-3\tilde{a}^{2}+f\big{)}$
(98) $\displaystyle w_{33}$
$\displaystyle=\frac{\tilde{A}_{0}}{\tilde{p}^{3}}f^{3/2}r^{2}$ (99)

If we return to (95) and we use $w_{00}$ and $w_{33}$, we get:

$r^{2}\phi^{\prime
2}+\kappa\frac{\tilde{A}_{0}}{\tilde{p}^{3}}f^{3/2}r^{2}\phi^{\prime
2}=f-\kappa\frac{\tilde{A}_{0}}{\tilde{p}^{3}}f^{3/2}(f-3\tilde{a}^{2})$ (100)

From (100) we find:

$\tilde{p}^{5}+4\kappa\tilde{A}_{0}f^{3/2}\tilde{p}^{2}-2\kappa\tilde{A}_{0}f^{5/2}=0$
(101)

In order to determine the radius of the photonsphere, we need two equations.
The first one is (101) and we find the second from the path equations. We get
the radial path equation by substituting $\mu=1$ in (IV) and if we use our
assumptions $r=$const. and $\theta=\frac{\pi}{2}$ we find:

$\displaystyle\frac{f(1-f)}{2r}\dot{t}^{2}-rf\dot{\phi}^{2}=$
$\displaystyle-\lambda\tilde{A}_{0}\left[\left(\frac{1}{2}\tilde{a}f^{1/2}\dot{t}-\frac{1}{4\tilde{a}}f^{3/2}\dot{t}^{3}\right)\frac{1-f}{r}+\frac{1}{2\tilde{a}}f^{3/2}r\dot{t}\dot{\phi}^{2}\right]$
(102)

where $\lambda=\frac{z}{1+z}$.
Then, by using (VII) and after some calculations we find:

$4f^{1/2}\tilde{p}^{3}+2\lambda\tilde{A}_{0}(1-2f)\tilde{p}^{2}+2f^{1/2}(1-3f)\tilde{p}-\lambda\tilde{A}_{0}f(1-3f)=0$
(103)

Therefore, the equations we need to solve are (101) and (103). If we take
(101) and set $\mu=f^{-1/2}\tilde{p}$ we get:

$\mu^{5}+4\kappa\tilde{A}_{0}\mu^{2}-2\kappa\tilde{A}_{0}=0$ (104)

By giving values to the parameters $\kappa$ and $\tilde{A}_{0}$ we can solve
(104) numerically and determine the value of $\mu$. Then, from the definition
of $\mu$ we can find a relation between $f$ and $\tilde{p}$ which can be
substituted in (103) to find the term $f$ and from this the radius of the
photonsphere. The results for different values of the parameters are shown on
the table that follows:

($a$ , $b$ , $c$) | $\tilde{A}_{0}$ | $\mu$ | $r/R_{s}$
---|---|---|---
$(1,1,1)$ | $10^{-3}$ | $0,25854$ | $1,53577$
$(1,1,1)$ | $10^{-4}$ | $0,16599$ | $1,51416$
$(1,1,1)$ | $10^{-6}$ | $0,06671$ | $1,50224$
$(1000,1,1)$ | $10^{-4}$ | $0,05245$ | $1,50138$
$(1,1000,1)$ | $10^{-4}$ | $0,17961$ | $1,51668$
$(1,1,1000)$ | $10^{-4}$ | $0,17961$ | $1,51667$
$(1,1000,1000)$ | $10^{-4}$ | $0,17963$ | $1,51668$

where $a,b,c$ are the starting parameters in the Lagrangian in (67) and
through them we calculate the term $\kappa=\frac{b+c}{a+b+c}$.

##  VIII Concluding remarks

In this article, we investigate further properties and applications of our
previous work of the SFR model which generalizes the classical Schwarzschild
spacetime by introducing a timelike covector $A_{\gamma}$ in the metric
structure [40]. This covector is specified by the solution of the generalized
Einstein equations of the SFR model. It provides the local anisotropy and may
cause Lorentz violating effects.

In addition, we derive the form of S-anisotropic curvature which takes a
geometrical meaning because of its dependence on coordinates.

The generalized Kretschmann-like curvature invariant plays a crucial role in
our approach since the horizontal $K_{H}$, rel.(65), coincides with the
Kretschmann invariant of the classical Schwarzschild solution which gives a
singularity at the point $r=0$. The second Kretschmann curvature invariant
$K_{V}$, rel.(66), provides information for singularities with more degrees of
freedom as we show and it is characterized by the scalar curvature S,
rel.(63).

In the framework of applications of SFR model we extend our study of timelike
geodesic paths and we compare them with corresponding paths of GR. We notice
that the extra terms in rel.(IV) act as an extra force that influences the
gravitational field and give a small deviation from the paths of GR.

In the last sections, we find the form of momentum and the energy in our
approach, rel.(81) and (82).

By considering the Lagrangian function (rel.(67)) we calculate the
gravitational redshift and the photonsphere for our case. While in the
redshift calculation we find no deviation from general relativity, in the
study of the photonsphere we find infinitesimal deviations from GR which may
be ought to the small anisotropic perturbations coming from Lorentz violation
effects.

##  Appendix A Variational principle on a Hilbert-like action

In this section, we present the basic steps of the variation of the action
(46):

$K=\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{R}+2\kappa\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}$
(105)

with respect to $g_{\mu\nu}$, $v_{\alpha\beta}$ and $N^{\alpha}_{\kappa}$ in
order to acquire the generalized field equations (48)-(50), see [44] for the
original derivation. Variating the total action, we get:

$\displaystyle\Delta K=$
$\displaystyle\,\int_{\mathcal{N}}d^{8}\mathcal{U}(R+S)\Delta\sqrt{|\mathcal{G}|}+\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}(\Delta
R+\Delta S)$
$\displaystyle+\,2\kappa\int_{\mathcal{N}}d^{8}\mathcal{U}\,\Delta\\!\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)$
(106)

with

$\displaystyle\Delta\sqrt{|\mathcal{G}|}=$
$\displaystyle-\frac{1}{2}\sqrt{|\mathcal{G}|}\left(g_{\mu\nu}\Delta
g^{\mu\nu}+v_{\alpha\beta}\Delta v^{\alpha\beta}\right)$ (107)
$\displaystyle\Delta R=$
$\displaystyle\,2g^{\mu[\kappa}\dot{\partial}_{\alpha}L^{\nu]}_{\mu\nu}\Delta
N^{\alpha}_{\kappa}+\overline{R}_{\mu\nu}\Delta
g^{\mu\nu}+\mathcal{D}_{\kappa}Z^{\kappa}$ (108) $\displaystyle\Delta S=$
$\displaystyle\,S_{\alpha\beta}\Delta
v^{\alpha\beta}+\mathcal{D}_{\gamma}B^{\gamma}$ (109)

where
$\overline{R}_{\mu\nu}=R_{(\mu\nu)}+\Omega^{\alpha}_{\kappa(\mu}C^{\kappa}_{\nu)\alpha}$
and

$\displaystyle Z^{\kappa}=$ $\displaystyle\,g^{\mu\nu}\Delta
L^{\kappa}_{\mu\nu}-g^{\mu\kappa}\Delta L^{\nu}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\,-\mathcal{D}_{\nu}\Delta
g^{\nu\kappa}+g^{\kappa\lambda}g_{\mu\nu}\mathcal{D}_{\lambda}\Delta
g^{\mu\nu}$
$\displaystyle+\,2\left(g^{\kappa\mu}C^{\lambda}_{\lambda\alpha}-g^{\kappa\lambda}C^{\mu}_{\lambda\alpha}\right)\Delta
N^{\alpha}_{\mu}$ (110) $\displaystyle B^{\gamma}=$
$\displaystyle\,v^{\alpha\beta}\Delta
C^{\gamma}_{\alpha\beta}-v^{\alpha\gamma}\Delta C^{\beta}_{\alpha\beta}$
$\displaystyle=$ $\displaystyle\,-\mathcal{D}_{\alpha}\Delta
v^{\alpha\gamma}+v^{\gamma\delta}v_{\alpha\beta}\mathcal{D}_{\delta}\Delta
v^{\alpha\beta}$ (111)

Stokes theorem on the Lorentz tangent bundle reads:

$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\mu}H^{\mu}=$
$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{T}^{\alpha}_{\mu\alpha}H^{\mu}$
$\displaystyle+\,\int_{\partial{\mathcal{N}}}n_{\mu}H^{\mu}\mathcal{\tilde{E}}$
(112)
$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\alpha}W^{\alpha}=$
$\displaystyle-\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,C^{\mu}_{\mu\alpha}W^{\alpha}$
$\displaystyle+\,\int_{\partial{\mathcal{N}}}n_{\alpha}W^{\alpha}\mathcal{\tilde{E}}$
(113)

where $H=H^{\mu}\delta_{\mu}$ and $W=W^{\alpha}\dot{\partial}_{\alpha}$ are
vector fields on $TM$, $\tilde{\mathcal{E}}$ is the Levi-Civita tensor on the
boundary $\partial{\mathcal{N}}$, $(n_{\mu},n_{\alpha})$ is the normal vector
on the boundary and
$\mathcal{T}^{\alpha}_{\mu\beta}=\dot{\partial}_{\beta}N^{\alpha}_{\mu}-L^{\alpha}_{\beta\mu}$.
Using relation (112) and eliminating boundary terms, we get

$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\kappa}Z^{\kappa}=\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{T}^{\alpha}_{\kappa\alpha}Z^{\kappa}$
$\displaystyle=\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\nu}\left[\mathcal{T}^{\beta}_{\kappa\beta}\left(-\Delta
g^{\nu\kappa}+g^{\nu\kappa}g_{\mu\lambda}\Delta g^{\mu\lambda}\right)\right]$
$\displaystyle\,-\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\left[-\mathcal{D}_{\nu}\mathcal{T}^{\beta}_{\mu\beta}+g_{\mu\nu}\mathcal{D}^{\lambda}\mathcal{T}^{\beta}_{\lambda\beta}\right]\Delta
g^{\mu\nu}$
$\displaystyle\,+2\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{T}^{\beta}_{\kappa\beta}\left(g^{\kappa\mu}C^{\lambda}_{\lambda\alpha}-g^{\kappa\lambda}C^{\mu}_{\lambda\alpha}\right)\Delta
N^{\alpha}_{\mu}$ (114)

where we have used the Leibniz rule for the covariant derivative. Using (112)
again and eliminating the new boundary terms, we get

$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\kappa}Z^{\kappa}=$
$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\left(\delta^{(\lambda}_{\nu}\delta^{\kappa)}_{\mu}-g^{\kappa\lambda}g_{\mu\nu}\right)\left(\mathcal{D}_{\kappa}\mathcal{T}^{\beta}_{\lambda\beta}-\mathcal{T}^{\gamma}_{\kappa\gamma}\mathcal{T}^{\beta}_{\lambda\beta}\right)\Delta
g^{\mu\nu}$
$\displaystyle+\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,4\mathcal{T}^{\beta}_{\kappa\beta}g^{\kappa[\mu}C^{\lambda]}_{\lambda\alpha}\Delta
N^{\alpha}_{\mu}$ (115)

Similarly, using relation (113) and eliminating the boundary terms, we get

$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\alpha}B^{\alpha}=-\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,C^{\mu}_{\mu\beta}B^{\beta}$
$\displaystyle=-\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\alpha}\left[C^{\mu}_{\mu\beta}\Delta
v^{\alpha\beta}-v^{\alpha\beta}v_{\gamma\delta}C^{\mu}_{\mu\beta}\Delta
v^{\gamma\delta}\right]$
$\displaystyle-\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\left(\mathcal{D}_{\alpha}C^{\mu}_{\mu\beta}-v^{\gamma\delta}v_{\alpha\beta}\mathcal{D}_{\gamma}C^{\mu}_{\mu\delta}\right)\Delta
v^{\alpha\beta}$ (116)

where again we used the Leibniz rule. Applying (113) again and eliminating the
new boundary terms, we get

$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\mathcal{D}_{\alpha}B^{\alpha}$
$\displaystyle=\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\,\left(v^{\gamma\delta}v_{\alpha\beta}-\delta^{(\gamma}_{\alpha}\delta^{\delta)}_{\beta}\right)\left(\mathcal{D}_{\gamma}C^{\mu}_{\mu\delta}-C^{\nu}_{\nu\gamma}C^{\mu}_{\mu\delta}\right)$
(117)

The matter part of the action is written as:

$\displaystyle\int_{\mathcal{N}}d^{8}\mathcal{U}\,\Delta\\!\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)$
$\displaystyle=$
$\displaystyle\,\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\frac{1}{\sqrt{|\mathcal{G}|}}\frac{\Delta\\!\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)}{\Delta
g^{\mu\nu}}\Delta g^{\mu\nu}$
$\displaystyle+\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\frac{1}{\sqrt{|\mathcal{G}|}}\frac{\Delta\\!\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)}{\Delta
v^{\alpha\beta}}\Delta v^{\alpha\beta}$
$\displaystyle+\int_{\mathcal{N}}d^{8}\mathcal{U}\sqrt{|\mathcal{G}|}\frac{1}{\sqrt{|\mathcal{G}|}}\frac{\Delta\\!\left(\sqrt{|\mathcal{G}|}\,\mathcal{L}_{M}\right)}{\Delta
N^{\alpha}_{\kappa}}\Delta N^{\alpha}_{\kappa}$ (118)

Finally, combining equations (106)-(111), (115), (117), (118) and setting
$\Delta K=0$, we get the equations (48)-(50) and the energy-momentum tensors
(51)-(53).

## Acknowledgements

The authors would like thank the unknown referee for his valuable comments and
remarks.

This research is co-financed by Greece and the European Union (European Social
Fund-ESF) through the Operational Programme “Human Resources Development,
Education and Lifelong Learning” in the context of the project “Strengthening
Human Resources Research Potential via Doctorate Research” (MIS-5000432),
implemented by the State Scholarships Foundation (IKY).

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# Girth Alternative for HNN Extensions

Azer Akhmedov, Pratyush Mishra  Department of Mathematics
North Dakota State University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove the Girth Alternative for a sub-class of the HNN extensions of
finitely generated groups. We also produce counterexamples to show that beyond
our class, the alternative fails in general.

##  1\. introduction

Let $G=\langle S|R\rangle$ be a finitely generated group and let $A$ and $B$
be two subgroups of $G$ with isomorphism $\phi:A\to B$. The HNN extension
group of G relative to subgroups A and B with stable letter $t$, denoted by
$(G,A,B,t)$ is the extended group containing $G$ defined as

$G^{\ast}_{\phi}=(G,A,B,t)=\langle G,t|t^{-1}at=\phi(a)\hskip
2.84544pt\text{for}\hskip 2.84544pta\in A\rangle$

where $A$ and $B$ are not only isomorphic but also conjugate via the map
$\phi$. The HNN extensions originally arised in topology as the fundamental
groups of a topological space when two subspaces are glued along a
homeomorphism. 111In this case one obtains a somewhat more general notion
where $A$ and $B$ are not necessarily subgroups of $G$ but have homomorphic
images in $G$. In recent decades, HNN extensions have been used as a popular
tool to construct examples and counterexamples of groups for questions in
combinatorial group theory.

In [Sc], Schleimer defined the girth of $G$ with respect to a finite
generating set $S$, denoted by $girth(G,S)$ as the length of the shortest non-
trivial relation in $G$ with respect to generating set $S$. And the girth of
the group $G$ is defined as

$girth(G)=\sup_{S\subset G}\\{girth(G,S)|\langle S\rangle=G,|S|<\infty\\}$

For a given group $G$, it is natural to ask whether the $girth(G)$ is finite
or infinite. In [A1], it is shown that finitely generated groups satisfying a
law, which are not isomorphic to $\mathbb{Z}$, have finite girth. Moreover, in
[A2], the author shows that for many classes of groups (word hyperbolic, one-
relator, linear groups not isomorphic to $\mathbb{Z}$), the property of having
infinite girth coincides with the property of containing a non-abelian free
subgroup and introduced the notion of Girth Alternative similar in spirit to
the well-known Tits Alternative. For a given class $\mathcal{C}$ of finitely
generated groups, $\mathcal{C}$ is said to satisfy the Girth Alternative if
any group from the class $\mathcal{C}$ has either infinite girth or is
virtually solvable.

In [A2], [A3], Akhmedov has proved the Girth Alternative for the class of
hyperbolic, linear, one-relator and $PL_{+}(I)$ groups. In [Ya], Yamagata
proves the Girth Alternative for convergence groups and irreducible subgroups
of the mapping class groups. Independently in [Nak], Nakamura proves the
Alternative for all subgroups of mapping class groups and also for the
subgroups of Out$(\mathbb{F}_{n})$ containing the irreducible elements having
irreducible powers.

The table below shows a dichotomy between Girth Alternative and Tits
Alternative for some classes of finitely generated groups that we are
interested in.

Groups |  Tits Alternative |  Girth Alternative
---|---|---
PL+(I) |  fails (Thompson’s group F) |  holds
Linear |  holds |  holds
1-relator |  holds |  holds
Hyperbolic |  holds |  holds
Homeo${}_{+}(I)$ |  fails (Thompson’s group F) |  fails (we prove in [AM])
Diff${}^{\omega}_{+}(I)$ |  unknown |  holds (we prove in [AM])
Residually finite |  fails |  fails
Group of formal power series (over field $\mathbf{k}$) |  fails for char$\mathbf{k}>0$ & unknown for char$\mathbf{k}=0$ |  fails for char$\mathbf{k}>0$ & holds for char$\mathbf{k}$=0
HNN Extensions |  holds for proper extensions |  holds for proper extensions (we prove in this paper)

In [CM], the authors construct an example of a finitely generated residually
$p$-group which is not virtually solvable, but satisfies a law. This example
shows that Tits Alternative and Girth Alternative fail in the class of
residually finite groups. Moreover, since every countably based pro-$p$ group
embeds into the group of formal analytic power serieses over $\mathbb{F}_{p}$
[C], both alternatives fail in the latter group as well.

In this paper, our main result is the Girth Alternative for a sub-class of HNN
extensions, showing for these sub-class, again the property of having infinite
girth coincide with the property of containing a non-abelian free subgroup.

###  1.1. Conventions.

For a given group $G$, we say the HNN extension $(G,A,B,t)$ is

  1. 1.

Proper, when both the underlying subgroups $A$ and $B$ are proper in $G$.

  2. 2.

Semi-proper, when one of the subgroup is proper and the other is full group
$G$.

  3. 3.

Full, when both $A$ and $B$ are full group $G$. 222our terminology may differ
from the terminology of many other authors; for example, a semi-proper HNN
extension is often called (e.g. in [Bu1], [Bu2]) an acsenifng (or stritly
ascending) HNN extension in many sources; in [Bu2], “proper” is used for HNN
extensions which are either proper or semiproper in our terminology.

######  Theorem 1.1.

For $G$ be a finitely generated group with $A$ and $B$ two proper subgroups
then $girth(\Gamma)=\infty$, where $\Gamma=(G,A,B,t)$ is a proper HNN
extension of $G$ relative to $A,B$ and $\phi$.

Since proper HNN extensions are never solvable (they contain a subgroup
isomorphic to $\mathbb{F}_{2}$), Theorem 1.1 implies Girth Alternative for
proper HNN extensions. However, the following result provides class of
counterexamples to show that beyond our sub-class as in Theorem 1.1, the
alternative fails in general.

######  Proposition 1.2.

For $G$ a finitely generated group satisfying a law with $A=G$ and $B$ a
proper subgroup of $G$, then $girth(\Gamma)<\infty$, where $\Gamma=(G,A,B,t)$
is a semi proper HNN extension relative to $G,B$ and $\phi$.

Treating the case of amalgamated free product of groups turns out to be easier
than HNN extensions; we provide a separate proposition devoted to this case.

######  Proposition 1.3.

Any proper amalgamated free product $A*_{C}B$, where $A,B$ are finitely
generated groups and $\max\\{A:C,B:C\\}\geq 3$ , has infinite girth.

We call an amalagamted free product proper if $C$ is a proper subgroup in both
$A$ and $B$. Notice that without the condition $\max\\{A:C,B:C\\}\geq 3$ the
claim does not hold since $girth(D_{\infty})=2<\infty$. 333We use the notation
$D_{\infty}$ for the infinite dihedral group $\langle a,b\ |\
a^{2}=b^{2}=1\rangle$. It is an infinite virtually cyclic group. Recall that a
finite dihedral group $D_{n},n\geq 2$ is given by the presentation $\langle
a,b\ |\ a^{2}=b^{2}=(ab)^{n}=1\rangle$. The term dihedral group will refier to
$D_{q}$ where $q\in(\mathbb{N}\backslash\\{0,1\\})\cup\ \\{\infty]$.

Interestingly, we also obtain the following result which shows that the Girth
Alternative holds in general for the class of any HNN extension (proper, semi
proper or full) of the non-abelian free group $F_{n}$ for $n\geq 2$.

######  Proposition 1.4.

Any HNN extension of the non-abelian free group $\mathbb{F}_{n}$ for any
$n\geq 2$ has infinite girth.

In the case of $n=1$, Proposition 1.4 easily fails even for semi-proper HNN
extensions since the Baumslag-Solitar group $BS(1,m)=\langle a,b\
|aba^{-1}=b^{m}\rangle$ is solvable hence has finite girth. On the other hand,
proper HNN extensions of $\mathbb{Z}$ are all non-solvable one-relator groups,
hence, by Theorem 3.1 in [A2], have infinite girth. Let us also point out that
HNN extension of a free group $\mathbb{F}_{k}$ are not necessarily linear; for
$k=1$, recall that the groups $BS(n,m)=\langle a,b\
|ab^{n}a^{-1}=b^{m}\rangle$ are non-Hopfian hence not linear for all
$n,m>1,n\neq m$, and for $k\geq 2$, examples are provided in [SD]

##  2\. Preliminary results

First, we prove the following proposition which is interesting also from a
combinatorial point of view.

######  Proposition 2.1.

Let $G$ be a non-cyclic group such that no quotient of $G$ is isomorphic to a
dihedral group $D_{n},n\in(\mathbb{N}\backslash\\{0,1\\})\cup\\{\infty\\}$,
and let $A,B$ be proper isomorphic subgroups of $G$. Then $G$ admits a finite
generating set $S$ such that $S\cap(A\cup B)=\emptyset$.

###### Proof.

Let $d=d(G)$ be the minimal cardinality of a generating set of $G$ and

$\mathcal{S}=\\{S\subset G||S|=d,\langle S\rangle=G\\}.$

Since $G$ is non-cyclic, we have $d\geq 2$. We introduce the following
quantities:

$\alpha(S)=|S\cap(A\backslash B)|,\beta(S)=|S\cap(B\backslash A)|$
$\gamma(S)=|S\cap(A\cap B)|,\delta(S)=|S\backslash(A\cup B)|$

We now claim that there exists a finite generating set $S\in\mathcal{S}$ such
that $\delta(S)=d$ (i.e. $\alpha(S)=0,\beta(S)=0,\gamma(S)=0$). Indeed, let
$S\in\mathcal{S}$ such that $\delta(S)$ is maximal. Assume $\delta(S)<d$.

Claim 1: $\alpha(S)=0$ or $\beta(S)=0.$

Proof: Indeed, assume that $\alpha(S)\geq 1$ and $\beta(S)\geq 1,$ with
$s_{1}\in S\cap(A\backslash B)$ and $s_{2}\in S\cap(B\backslash A)$. Then
replace $S$ with $S^{\prime}=(S\backslash\\{s_{1}\\})\cup\\{s_{1}s_{2}\\}$.
Since $s_{1}s_{2}\not\in A\cup B$, we obtain that
$\delta(S^{\prime})=\delta(S)+1$, contradicting maximality of $\delta(S)$.

Thus, without loss of generality, we may and will assume that $\beta(S)=0$.
Notice that $\delta(S)>0$ because $A$ is a proper subgroup of $G$.

Claim 2: $\alpha(S)+\gamma(S)\leq 1$

For $S$ as in Claim 1, suppose $\alpha(S)+\gamma(S)>1$, let $s_{1},s_{2}\in
S\cap A,s_{1}\neq s_{2}$. Let also $s_{3}\in S\backslash(A\cup B)$, then
$s_{1}s_{3}\not\in A$ but

$s_{1}s_{3}\not\in A\implies s_{1}s_{3}\in B$

Replace $S$ by $S^{\prime}=\\{s_{1}s_{3},s_{2},s_{3},\dots,s_{n}\\}$, note
that as $\beta(S^{\prime})=1\neq 0$ then by Claim 1 we have
$\alpha(S^{\prime})=0$, which forces $s_{2}\in S\cap(A\cap B)$. Similarly, let
$S^{\prime\prime}=\\{s_{2}s_{3},s_{1},s_{3},\dots,s_{n}\\}$ and symmetrically,
we obtain that $s_{1}\in S\cap(A\cap B)$. Then $s_{1}s_{3}\notin B$.
Contradiction.

Claim 3: $\alpha(S)+\gamma(S)=0$

Proof: Assuming the contrary, let $\alpha(S)+\gamma(S)=1$ by Claim 2. Hence
$|S\cap A|=1$, so let $S\cap A=\\{a\\}$ and $S\backslash
A=\\{s_{1},\dots,s_{d-1}\\}$. Notice that for all $1\leq i\leq d-1$,

$as_{i}\not\in A,as_{i}\in B$

Choose $s_{j}\in S$ such that $s_{j}\not\in B$ (such an $s_{j}$ exists
otherwise that would lead to $\delta(S)=0$). Then, for all $1\leq i\leq d-1$,

$as_{i}s_{j}\in A,as_{i}s_{j}\not\in B$

So, using maximality of $\delta(S),$ inductively on the length of $m\geq 1$ of
a reduced word $w=w(s_{1},s_{2},\dots,s_{d-1})$ in the alphabet set
$\\{s_{1}^{\pm 1},\dots,s_{d-1}^{\pm 1}\\}$, it follows that

$aw\not\in A,aw\in B\hskip 2.84544pt\text{if $m$ is odd},$ $aw\in A,aw\not\in
B\hskip 2.84544pt\text{if $m$ is even}.$

Now, let $\mathcal{W}$ be the set of reduced words in the alphabet
$\\{s_{1}^{\pm 1},\dots,s_{d-1}^{\pm 1}\\},$

$H=\\{g\in G|g=w(s_{1},\dots,s_{d-1})\hskip 2.84544pt\text{such that}\hskip
2.84544ptw\in\mathcal{W}\\},$ $H_{1}=\\{g\in H|g=w(s_{1},\dots,s_{d-1})\hskip
2.84544pt\text{can be written as a word of even length in $H$ in}\hskip
2.84544pts_{1}^{\pm 1},\dots,s_{d-1}^{\pm 1}\\}$

Then, $[H:H_{1}]\leq 2$, hence $H_{1}\trianglelefteq H$. Also,
$aH_{1}\subseteq A$, but note that for all $i\in\mathbb{Z}$,

(1) $aH_{1}\subseteq A\implies H_{1}\subseteq A\implies
a^{i}H_{1}a^{-i}\subseteq A$

Then as, $H_{1}\trianglelefteq H$, we also have for all $1\leq i\leq d-1$,

(2) $s_{i}H_{1}s_{i}^{-1}\subseteq H_{1}\subseteq A$

Any word in $H_{1}$ can be written as product of odd words $u^{-1},v\in H$ and
we know that $au,av\in B$ so we have,

(3) $u^{-1}v=u^{-1}a^{-1}av=(au)^{-1}av\in B\implies H_{1}\subseteq B$

Similarly, $aH_{1}a^{-1}\subseteq B$ and for any $u$ odd word in $H$

(4) $a^{2}=auu^{-1}a=au(a^{-1}u)^{-1}\in B$

From $(1),(2),(3)$ and $(4)$ we get

$N_{G}(H_{1},a^{2})\subseteq A\cap B$

Similarly, for all $1\leq i\leq n,\epsilon\in\\{-1,1\\}$ we can write

$(as_{i}^{\epsilon})^{2}=as_{i}^{\epsilon}uu^{-1}as_{i}^{\epsilon}=(as_{i}u)u^{-1}(as_{i}^{\epsilon}u^{-1})u$

where $u$ is an odd word in $H$, thus we obtain that
$(as_{i}^{\epsilon})^{2}\in N_{G}(H_{1},a^{2})$.

Also, in the quotient $G/N_{G}(H_{1},a^{2})$ we have
$\overline{s_{i}}=\overline{s_{j}},1\leq i,j\leq d-1$ (i.e. the images of
$s_{i}$ and $s_{j}$ are equal) thus by taking $b=as_{1}$ we obtain that this
quotient is isomorphic to a quotient of infinite dihedral group,

$D_{\infty}=\langle\bar{a},\bar{b}|\bar{a}^{2}=\bar{b}^{2}=e\rangle$

where $N_{G}(H_{1},a^{2})$ is the normal closure of $\\{H_{1},a^{2}\\}$ in
$G$. So, we get a quotient of $G$ which is isomorphic to a quotient of
$D_{\infty}$ but any such quotient is isomorphic to
$D_{n},n\in(\mathbb{N}\backslash\\{0,1\\})\cup\\{\infty\\}$ which contradicts
our assumption. ∎

From the proof we immediately obtain the following

######  Proposition 2.2.

Let $G$ be a finitely generated group, $A,B$ be proper subgroups of $G$. If
there exists no finite generating set $S$ of $G$ with $S\cap(A\cup
B)=\emptyset$, then $G$ admits a normal subgroup $N\triangleleft G$ such that
$N\leq A\cap B$, and $N$ has index at most two in both $A$ and $B$.
Moreover,if $A,B$ are isomorphic and $(G,A,B,t)$ is a proper HNN extension,
then the latter has a quotient isomorphic to a proper HNN extension
$(D,\overline{A},\overline{B},t)$ where either $D=D_{\infty}=\langle a,b\ |\
a^{2}=b^{2}=1\rangle$ or $D=D_{n}=\langle a,b\ |\
a^{2}=b^{2}=(ab)^{n}=1\rangle,n\geq 2$, and $\overline{A}=\langle
a\rangle,\overline{B}=\langle b\rangle$.

Indeed, it suffices to take $N=N_{G}(H_{1},a^{2})$ from the proof of
Proposition 2.1. Notice that $N$ is a normal closure of words of even length
(in particular, it is a normal closure of
$\\{a^{2}\\}\cup\\{(as_{i}^{\epsilon})^{2}:1\leq i\leq
d-1,\epsilon\in\\{-1,1\\}\\}\cup\\{s_{i}^{\epsilon}s_{j}^{\delta}:1\leq
i,j\leq d-1,\epsilon,\delta\in\\{-1,1\\}\\}$) and $N\subseteq A\cap B$ thus
$t^{-1}at\notin N$. So we can take $b=t^{-1}at$. Then the images of $a,b$ in
the quotient $G/N$ will generate a dihedral group.

##  3\. Proof of Theorem 1.1

First, we prove the following claim which shows the use of Proposition 2.1.

######  Proposition 3.1.

Let $G$ be a group with a finite generating set $S$, and
$G^{\ast}_{\phi}=(G,A,B,t)$ be an HNN extension where $A,B$ are proper
subgroups and $(A\cup B)\cap S=\emptyset$. Then
$girth(G_{\phi}^{\ast})=\infty$.

###### Proof.

The proof is a direct application of Britton’s Lemma [Brit]. 555Britton’s
Lemma states that in an HNN extension $(G,A,B,t)$, if a word $w$ can be
expressed $w=g_{0}t^{\epsilon_{1}}g_{1}\dots...t^{\epsilon_{1}}g_{n},n\geq 1$,
with no subwords of the form $t^{-1}g_{i}t,g_{i}\in A$, or
$tg_{j}t^{-1},g_{j}\in B$, then $w\neq 1$. Letting $S=\\{s_{1},\dots,s_{n}\\}$
where $1\notin S$ for any $r\geq 1$, we can take
$S^{(r)}=\\{t,t^{r}s_{1}t^{-r},t^{2r}s_{2}t^{-2r},\dots,t^{nr}s_{n}t^{-nr}\\}$.
By Britton’s Lemma, $girth(G^{\ast}_{\phi},S^{(r)})\geq r$ and since $r\geq 1$
is arbitrary, we conclude that $girth(G^{\ast}_{\phi})=\infty$. ∎

We now prove separately the fact that a proper HNN extension of $D_{\infty}$
has infinite girth. This covers the case when we are unable to separate the
generating set $S$ from the union $A\cup B$.

######  Proposition 3.2.

Let $D_{q},q\in(\mathbb{N}\backslash\\{0,1\\})\cup\\{\infty\\}$ be a dihedral
group with standard generators $a,b$, $A$ and $B$ be two proper isomorphic
subgroups. Then, $girth((D_{q},A,B,t))=\infty$ for all
$q\in(\mathbb{N}\backslash\\{0,1\\})\cup\\{\infty\\}$ .

###### Proof.

We break the proof into the following two cases:

Case 1: If $A$ and $B$ are proper cyclic isomorphic subgroups of $D_{q}$, then
the hypothesis of Proposition 3.1 can still be arranged. It is easy to see
that any proper cyclic subgroup of $D_{q}$ is isomorphic either to a cyclic
group of size $q$ or to $\mathbb{Z}/2\mathbb{Z}$. Let $A=\langle
w_{1}\rangle;B=\langle w_{2}\rangle$ be two proper cyclic subgroups of
$D_{q}$. Then we have the following sub-cases:

Sub-case 1: For the word length $l(w_{1})>1,l(w_{2})>1$, $S=\\{a,b\\}$ works,
that is $S\cap(A\cup B)=\emptyset$.

Sub-case 2: For $l(w_{1})=1,l(w_{2})>1$, let $w_{1}=\\{a\\}$ then the
generating set $S=\\{b,ab\\}$ does not intersects $A$ and $B$.

Sub-case 3: For $l(w_{1})=1,l(w_{2})=1$. Choose $S=\\{aba,ab\\}$.

Hence, we see that the arrangements of Proposition 3.1 still happens in the
above sub-cases.

Case 2: $A$ and $B$ are non-cyclic isomorphic subgroups of $D_{q}$.

Any non-cyclic subgroup of $D_{\infty}$ is isomorphic to $D_{\infty}$; such a
subgroup will be of the form $G_{m,n}=\langle a(ba)^{m},b(ab)^{n}\rangle$
where $m,n\in\mathbb{N}\cup\\{0\\}$. Notice that $G_{0,0}=D_{\infty}$ and all
other subgroups are proper. Similarly, any non-cyclic subgroup of $D_{q}$ for
a finite $q\geq 2$ is isomorphic to $D_{q_{1}}$ where $q_{1}|q$, and these
subgroups are also of the form $G_{m,n}=\langle a(ba)^{m},b(ab)^{n}\rangle$
for some $m,n\geq 0$. Let $S=\\{a,b\\},A=G_{m,n},B=G_{k,l}$ where
$\max\\{m,n\\}\geq 1$ and $\max\\{k,l\\}\geq 1$.

If $A\cap S=B\cap S=\emptyset$, then the hypothesis of Proposition 2.1 is
satisfied hence $girth((D_{q},A,B,t))=\infty$.

If $A\cap S=\emptyset$ but $B\cap S\neq\emptyset$, then without loss of
generality we may assume $b\in B\cap S$. Then $a\notin S$ and replacing $S$
with $S^{\prime}=\\{a,ab\\}$ we again satisfy the hypothesis of Proposition
2.1.

The case of $B\cap S=\emptyset$ but $A\cap S\neq\emptyset$ is treated
similarly. Thus we are left the case when $A\cap S\neq\emptyset$ and $B\cap
S\neq\emptyset$. Then we may assume $a\in A,b\in B$ and
$t^{-1}at=u,tbt^{-1}=v$ where $u=a(ba)^{m},v=b(ab)^{n},m,n\geq 1$. For every
$r\geq 1$, we let $S^{(r)}=\\{t,t^{r}at^{-2r},t^{-r}bt^{2r}\\}$. We consider
words in the alphabet $t,X=t^{r}at^{-2r},Y=t^{-r}bt^{2r}$.

Notice that for all $n\geq 1$,

$X^{n}=t^{r}W_{1}t^{-(n+1)r},X^{-n}=t^{(n+1)r}W_{2}t^{-r},Y^{-n}=t^{r}W_{3}t^{-(n+1)r},Y^{n}=t^{(n+1)r}W_{4}t^{-r}$

where $W_{i},1\leq i\leq 4$ are suitable in the sense that it can be written
as $W_{i}=u_{1}t^{p_{1}}u_{2}t^{p_{2}}\dots u_{k}t^{p_{k}}u_{k+1}$ with
$p_{i},1\leq i\leq k$ being non-zero integers such that if $u_{i}\in A$, then
$p_{i-1}>0$ (if $i\geq 1),p_{i}<0$, and if $u_{i}\in B$, then $p_{i-1}<0$ (if
$i\geq 1),p_{i}>0$. Then any word of length less than $r$ in the alphabet
$\\{t,X,Y\\}$ will be still suitable, hence by Britton’s Lemma such a word is
not identity. ∎

The proof of Theorem 1.1 follows from Proposition 2.1, Proposition 2.2,
Proposition 3.1 and Proposition 3.2 with an application of Britton’s lemma. To
be more precise, we have the following cases:

Case 1: If there exists a quotient of $G$ isomorphic to a dihedral group
$D_{n}$ i.e., $G/N\simeq D_{n}$ for some
$n\in(\mathbb{N}\backslash\\{0,1\\})\cup\\{\infty\\}$ then again we have the
following sub-cases:

Sub-case 1: If the projection subgroups $\bar{A}$ and $\bar{B}$ in $G/N\simeq
D_{n}$ are proper isomorphic cyclic subgroups in $G/N$ then as shown in Case 1
in Proposition 3.2, there exists $\bar{S}\subset G/N$ such that
$\bar{S}\cap(\bar{A}\cap\bar{B})=\emptyset$, which implies $S\cap(A\cap
B)=\emptyset$ in $G$, satisfying the hypothesis of Proposition 2.1 and then by
Proposition 3.1, the proof follows immediately for this sub-case.

Sub-case 2: If the projection subgroups $\bar{A}$ and $\bar{B}$ in $G/N\simeq
D_{n}$ are proper non cyclic isomorphic subgroups of $G/N$, then we already
proved this sub-case in Case 2 of Proposition 3.2

Case 2: If there exists no quotient of $G$ isomorphic to a dihedral group
$D_{n}$ for any $n\in(\mathbb{N}\backslash\\{0,1\\})\cup\\{\infty\\}$, then
the proof for this case follows immediately by Proposition 2.1, Proposition
2.2 and Proposition 3.1.

This completes the proof of Theorem 1.1

##  4\. Proof of Proposition 1.2

Given a semi-proper HNN extension $\Gamma=(G,A,B,t)$ with $A=G$, we can form a
union
$\mathcal{G}=\displaystyle\mathop{\bigcup}_{n\in\mathbb{Z}}t^{n}Gt^{-n}$.
Notice that, since $tGt^{-1}=B$, we will have a two-sided infinite chain

$\dots<t^{-2}Gt^{2}<t^{-1}Gt<G<tGt^{-1}<t^{2}Gt^{-2}<\dots$

of strict inclusions. Then $\mathcal{G}$ is a normal subgroup and
$\Gamma/\mathcal{G}\cong\mathbb{Z}$.

Now, let $G$ be a group satisfying a law. Then $G$ satisfies a law $W(x,y)$ in
two variables. To show the claim of Proposition 1.2, we just need to observe
that then for all $n\in\mathbb{Z}$, $t^{n}Gt^{-n}$ also satisfies the law
$W(x,y)$ hence by the strict inclusions, the normal subgroup $\mathcal{G}$
satisfies $W(x,y)$ as well. On the other hand, the infinite cyclic group
$\mathbb{Z}$ also satisfies a law (e.g. $[x,y]=1$ in $\mathbb{Z}$). It remains
to notice that for any short exact sequence $1\to L\to H\to K\to 1$ of groups,
$H$ satisfies a law iff $L$ and $K$ satisfy a law. This finishes the proof of
Proposition 1.2.

In the above proof we indeed realized the HNN extension $(G,A,B,t)$ as a semi-
direct product $\mathbb{Z}\ltimes_{t}\mathcal{G}$. The normal subgroup
$\mathcal{G}$ is a particularly meaningful object in the case when $G$ is a
finitely-generated nilpotent group. For example, if $G$ is also torsion-free,
then $\mathcal{G}$ naturally lies inside the Malcev completion of $G$ (see
[Mal]) and one can also treat a general case of a finitely-generated nilpotent
group with possibly some torsions.

Let us recall that torsion elements of a nilpotent group $G$ form a subgroup,
called the torsion subgroup. We will write $Tor(G)$ to denote the torsion
subgroup. In addition, if $G$ is finitely generated then the torsion subgroup
$Tor(G)$ is finite and normal, moreover, the quotient $G/Tor(G)$ is torsion-
free. Now, if we have isomorphic subgroups $A,B\leq G$ with $A=G$, then
necessarily $Tor(B)=Tor(A)=Tor(G)$. Hence, for a semi-proper HNN extension
$(G,A,B,t)$ where the conjugation by $t$ is given by an isomorphism $\phi:A\to
B$ (i.e. by $\phi:G\to B$) then $\phi|_{Tor(G)}:Tor(G)\to Tor(B)$ is an
isomorphism and we also obtain an induced isomorphism $\phi_{1}:G/Tor(G)\to
B/Tor(G)$. Then the group $(G,A,B,t)$ admits a normal subgroup
$\mathbb{Z}\ltimes_{\phi}Tor(G)$ whereas the quotient by this normal subgroup
is isomorphic to $(G/Tor(G),A/Tor(G),B/Tor(G),t_{1})$ where the conjugation by
$t_{1}$ is given by the isomorphism $\phi_{1}$.

Let $G_{1}=G/Tor(G),A_{1}=A/Tor(G),B_{1}=B/Tor(G)$. We consider the HNN
extension $(G_{1},A_{1},B_{1},t_{1})$ given by the isomorphism $\phi_{1}$.
Notice that this is a semi-proper HNN extension since $A_{1}=G_{1}$, moreover,
$G_{1}$ is torsion-free. Now we use the fact that a finitely generated
torsion-free nilpotent group $H$ admits a Malcev completion $\overline{H}$
which is also nilpotent of the same nilpotency degree, moreover, any
monomorphism $\psi:H\to H$ can be extended to an isomorphism
$\overline{\psi}:\overline{H}\to\overline{H}$. Thus the monomorphism
$\phi_{1}:G_{1}\to G_{1}$ can be extended to an isomorphism
$\overline{\phi_{1}}:\overline{G_{1}}\to\overline{G_{1}}$. Then the HNN
extension $(G_{1},A_{1},B_{1},t_{1})$ is a subgroup of
$(\overline{G_{1}},\overline{A_{1}},\overline{B_{1}},t_{2})$ where the
conjugation by $t_{2}$ is given by the isomorphism $\overline{\phi_{1}}$.
Hence the group $(\overline{G_{1}},\overline{A_{1}},\overline{B_{1}},t_{2})$
is a semidirect product
$\mathbb{Z}\ltimes_{\overline{\phi_{1}}}\overline{G_{1}}$. Thus
$(\overline{G_{1}},\overline{A_{1}},\overline{B_{1}},t_{2})$ is nilpotent.
Hence the original HNN extension $(G,A,B,t)$ is a subgroup of a nilpotent
extension of a nilpotent group. Since $(G,A,B,t)$ is not infinite cyclic, in
particular, we again see that it has a finite girth.

A great example of a Malcev completion can be described for an integral
Heisenberg group $H_{\mathbb{Z}}=<x,y|[[x,y],x]=[[x,y],y]=1>$. This group is
isomorphic to the group $U_{3}(\mathbb{Z})$ of integral unipotent matrices of
size $3\times 3$. The Malcev closure of $H_{\mathbb{Z}}$ will be equal to
$U_{3}(\mathbb{R})$, the group of real unipotent matrices of size $3\times 3$.
A semi-proper HNN extension of $H_{\mathbb{Z}}$ will be a subgroup of
$\mathbb{Z}\ltimes_{\phi}U_{3}(\mathbb{R})$.

However, note that for proper extensions of nilpotent groups this construction
fails. Indeed, the following proper HNN extension of $\mathbb{Z}^{2}=\langle
a,b\rangle$ has infinite girth, which is in support to our Theorem 1.1,
$(\mathbb{Z}^{2},\langle b^{-1}\rangle,\langle a^{n}b^{-1}\rangle,t)$ for any
$n\in\mathbb{Z}$ with $\phi$,

$\phi:\langle b^{-1}\rangle\to\langle a^{n}b^{-1}\rangle$

Then,

$(\mathbb{Z}^{2})^{\ast}_{\phi}=\langle
a,b,t|[a,b]=e,t^{-1}b^{-1}t=a^{n}b^{-1}\rangle=\langle
a,b,t|b^{-1}ab=a,b^{-1}tb=ta^{n}\rangle=\mathbb{F}_{2}\rtimes_{\phi}\mathbb{Z}$

But, it follows from Proposition 1.4 that

$girth(\mathbb{F}_{2}\rtimes_{\phi}\mathbb{Z})=\infty$

##  5\. Proof of Proposition 1.3

Given some groups $A,B,C$ with monomorphisms $\phi:C\to A$ and $\psi:C\to B$,
one can form a product of $A$ and $B$ amalgamated over $C$. We will write this
as $A*_{C}B$ dropping $\phi$ and $\psi$ from the notation as they will be
given to us in the context. It turns out that $A,B,C$ will have isomorphic
images in $G=A*_{C}B$ which we still denote with the same letters. We will be
using the following well known analog of Britton’s Lemma for amalgamated free
products: Let $T_{A}=A\backslash C,T_{B}=B\backslash C$ and
$w=g_{0}g_{1}\dots...g_{n},n\geq 1$ such that for all $1\leq i\leq n$ if

(i) if $g_{i-1}\in A$, then $g_{i}\in T_{B}$

(ii) if $g_{i-1}\in B$, then $g_{i}\in T_{A}$

(iii) $g_{0}\neq 1$

Then $w\neq 1\in A*_{C}B$.

We will call the amalgamated free product proper if $C$ is a proper subgroup
of both $A$ and $B$. Notice that if $C=A$ ($C=B$) then $G$ becomes isomorphic
to $B$ (to $A$) so the girth$(G)=\infty$ iff girth$(B)=\infty$
(girth$(A)=\infty$).

Let $G=A*_{C}B$ where $A,B$ are finitely generated groups. If $C$ is trivial,
then $G=A*B$ and in this simpler case we can procced as follows: Let
$S_{1}=\\{a_{1},\dots,a_{n}\\},S_{2}=\\{b_{1},\dots,b_{m}\\}$ be generating
sets of $A$ and $B$ respectively.where $1\notin S_{1},1\notin S_{2},S_{i}\cap
S_{i}^{-1}=\emptyset,1\leq i\leq 2$. If $A$ and $B$ are both cyclic groups
then the claim is an easy exercise (alternatively, the group $A*B$ is word
hyperbolic hence the result about the girth follows from Theorem 2.6 in [A2]),
so we will assume that at least one of them, say $A$, is not cyclic.Then
$n\geq 2$. Let now $r\geq 1$. We take reduced words
$U_{1}(X,Y,Z),\dots,U_{m+n}(X,Y,Z),V_{1}(X,Y,Z),\dots V_{m+n}(X,Y,Z)$ in the
free group formally generated by letters $X,Y,Z$ such that for all $1\leq
i,j\leq m+n$

(i) $U_{i}$ ends with $Y$ and $V_{j}$ starts with $Y^{-1}$;

(ii) $|U_{i}|=2p,|V_{j}|=2p,|U_{i}^{\epsilon}V_{j}^{\delta}|>\frac{3}{2}p$ for
all $\epsilon,\delta\in\\{-1,1\\}$ where $p>2r$;

(iii) $U_{i}$ and $V_{j}$ are reduced words of length at least $p$ in the
alphabet $\\{\xi,\eta\\}$ where $\xi=XY,\eta=ZY$.

Now we let $S^{(r)}=S^{(r)}_{1}\cup S^{(r)}_{2}\cup\\{uv,wv\\}$ where

$S^{(r)}_{1}=\\{U_{1}(u,v,w)a_{1}V_{1}(u,v,w),\dots,U_{n}(u,v,w)a_{n}V_{n}(u,v,w)\\}$

and

$S^{(r)}_{2}=\\{U_{n+1}(u,v,w)a_{1}b_{1}a_{1}V_{n+1}(u,v,w),\dots,U_{m}(u,v,w)a_{1}b_{m}a_{1}V_{m}(u,v,w)\\}$

where
$u=a_{1}b_{1}a_{1}^{-1},v=a_{2}b_{1}a_{2}^{-1},w=a_{1}a_{2}b_{1}a_{2}^{-1}a_{1}^{-1}$.

By condition (iii) we have that $U_{i},V_{i},1\leq i,j\leq m+n$ are also words
of length at least $p$ in $uv$ and $wv$. Then $S^{(r)}$ is a generator of
$G=A*_{C}B$. Notice that none of the words

$a_{1},a_{2},a_{1}^{-1}a_{2},a_{2}a_{1}^{-1},a_{1}^{-1}a_{2}a_{1},a_{1}a_{2}a_{1}^{-1},a_{2}^{-1}a_{1}a_{2},a_{2}a_{1}a_{2}^{-1}$

represents identity element in $A$, hence there is no relation of length less
than $r$ among the elements of $S^{(r)}$. Thus girth$(A*B,S^{(r)})\geq r$.

In the general case, when $C\neq 1$, by the index assmption of our
proposition, without loss of generality, we may assume that $A:C\geq 3$.

We will use the following simple

######  Lemma 5.1.

Let $G$ be a group and $H\leq G$ be a subgroup such that for all $x\in
G\backslash H,x^{2}\in H$. Then $H$ is a normal subgroup.

###### Proof.

Indeed, let $h\in H,x\in G\backslash H$. $xh\in G\backslash H$ hence
$(xh)^{2}\in H$ which yields $xhx\in H$. Then $xhx^{-1}=(xhx)x^{-2}\in H$. ∎

Using Lemma 5.1, we can claim another simple

######  Lemma 5.2.

Let $A$ be a group and $C\leq A$ with $A:C\geq 3$. Then there exists distinct
$a_{1},a_{2}\in A$ such that
$a_{1},a_{2},a_{1}^{-1}a_{2},a_{2}a_{1}^{-1}\notin C$ and
$a_{1}^{-1}a_{2}a_{1},a_{1}a_{2}a_{1}^{-1},a_{2}^{-1}a_{1}a_{2},a_{2}a_{1}a_{2}^{-1}\notin
C$.

###### Proof.

If $C$ is not trivial and there exists $x\in A\backslash C$ such that
$x^{2}\notin C$, then we can take $a_{1}=x,a_{2}=x^{2}$; otherwise, by Lemma
5.1, $H$ is a normal subgroup thus we just need to find two distinct elements
$d_{1},d_{2}\in(G/H)\backslash\\{1\\}$ satisfying conditions
$d_{1}^{-1}d_{2}\neq 1$ but such elements trivially exist if $|G/H|\geq 3$. ∎

Then we can choose generating sets
$S_{1}=\\{a_{1},\dots,a_{n}\\},S_{2}=\\{b_{1},\dots,b_{m}\\}$ of $A$ ad $B$
such that $S_{1}\cap C=S_{2}\cap C=\emptyset$, $n\geq 2$ and None of the
elements

$a_{1},a_{2},a_{1}^{-1}a_{2},a_{2}a_{1}^{-1},a_{1}^{-1}a_{2}a_{1},a_{1}a_{2}a_{1}^{-1},a_{2}^{-1}a_{1}a_{2},a_{2}a_{1}a_{2}^{-1}$

belong to $C$. Then we define $S^{(r)}$ as above and there will be no relation
of length less than $r$ among the elements of $S^{(r)}$. Thus we again obtain
that girth$(A*_{C}B,S^{(r)})\geq r$. Since $r$ is arbitrary, we conclude that
girth$(A*_{C}B)=\infty$.

######  Remark 5.3.

Notice that a proper amalgamated free product $G=A*_{C}B$ is virtually
solvable iff $C$ is a virtually solvable normal subgroup and $G/C\cong
D_{\infty}$ so Proposition 1.3 indeed confirms the Girth Alternative for the
class of proper amalgamated free products.

##  6\. Proof of Proposition 1.4

It suffices to prove the theorem for $\mathbb{F}_{2}=\langle x,y\rangle$ since
an HNN extension of $\mathbb{F}_{n}$ for $n\geq 3$ surjects onto the HNN
extension of $\mathbb{F}_{2}$ and infinite girth of latter implies infinite
girth of the former by Proposition 1.1 in [A2].

  1. 1.

Case 1: Proper HNN extension $(\mathbb{F}_{2},A,B,t)$, with $\phi:A\to B$ an
isomorphism between proper subgroups $A,B<\mathbb{F}_{2}$.

It follows from Theorem 1.1 that

$girth((\mathbb{F}_{2},A,B,t))=\infty$

  2. 2.

Case 2: Semi-proper HNN extension $(\mathbb{F}_{2},\mathbb{F}_{2},B,t)$, with
$B=\langle w_{1}(x,y),w_{2}(x,y)\rangle$ a proper subgroup of $\mathbb{F}_{2}$
with the isomorphism $\phi$ mapping

$x\mapsto w_{1}(x,y),y\mapsto w_{2}(x,y)$

Then, the HNN extension is given by:

$(\mathbb{F}_{2},\mathbb{F}_{2},B,t)=\langle
x,y,t|t^{-1}xt=w_{1}(x,y),t^{-1}yt=w_{2}(x,y)\rangle$

In order to prove Case 2, it suffices to prove the case when the word
$|w_{1}|>1$ and $|w_{2}|>1$ and $w_{1},w_{2}\neq x^{n},y^{n}$. As for any
other case, one can use elementary Nielson transformations to $w_{1}$ and
$w_{2}$ to make the word length $|w_{1}|,|w_{2}|>1$ with $w_{1},w_{2}\neq
x^{n},y^{n}$.

So, assume $|w_{1}|>1$ and $|w_{2}|>1$ and $w_{1},w_{2}\neq x^{n},y^{n}$.

Notice that for any positive integer $r>0$, there exists $k>0$, (basically any
$k>2r$ works) such that for $-r\leq i<0$ and $|j-k|<r$ we have by an
application of Britton’s lemma [Brit],

$\mathbb{F}_{3r-1}\simeq\langle t^{-i}xt^{i},t^{j}yt^{-j}\rangle$

Then, there exists no relation of length $\leq r$ among the elements of the
generating set $S^{(r)}=\\{t,t^{-i+r}xt^{i+r},t^{j+r}yt^{-j+r}\\}$ and hence,

$girth((\mathbb{F}_{2},\mathbb{F}_{2},B,t),S^{(r)})>r$

Hence, we conclude that $girth((\mathbb{F}_{2},\mathbb{F}_{2},B,t))=\infty$
for semi proper HNN extensions of $\mathbb{F}_{2}$.

  3. 3.

Case 3: Full HNN extension
$(\mathbb{F}_{2})^{\ast}_{\phi}=(\mathbb{F}_{2};\mathbb{F}_{2};\mathbb{F}_{2};t)=\mathbb{Z}\ltimes_{\phi}\mathbb{F}_{2},$
where $\phi$ is an automorphism of $\mathbb{F}_{2}$.

It is not known whether free-by-cyclic groups are linear, but this is known to
be true in the case when the rank of the free group is two [Bu3]. Thus our
group $\mathbb{Z}\ltimes_{\phi}\mathbb{F}_{2}$ is linear and it follows from
Theorem 4.4 in [A2] that
girth($\mathbb{Z}\ltimes_{\phi}\mathbb{F}_{2})=\infty$. This completes the
proof of Proposition 1.4

######  Remark 6.1.

Let us emphasize that free-by-cyclic groups
$\mathbb{Z}\ltimes_{\phi}\mathbb{F}_{n}$ are not always hyperbolic so we
cannot invoke the result from [A2] (Theorem 2.6) about hyperbolic groups. In
fact, by the result of P.Brinkmann [Brin] these groups are hyperbolic
precisely when they are atoroidal, i.e. when they do not contain an isomorphic
copy of $\mathbb{Z}\oplus\mathbb{Z}$. Moreover, for $n=2$,
$\mathbb{Z}\ltimes_{\phi}\mathbb{F}_{n}$ is always toroidal, hence not
hyperbolic.

The following general question regarding full HNN extensions of a given group
$G$ is also interesting to us:

Question 1: Let $G$ be a non-cyclic finitely generated group then, is it true
that $girth(G)=\infty$ if and only if
$girth(\mathbb{Z}\ltimes_{\phi}G)=\infty$.

One direction of the above question is easy, namely, if $girth(G)=\infty$,
then since $G$ is a quotient of $\mathbb{Z}\ltimes_{\phi}G$ by Proposition
1.1.in [A2], we obtain that $girth(\mathbb{Z}\ltimes_{\phi}G)=\infty$. We
conjecture that the other direction does not hold, i.e. there exists a
finitely generated group $G$ and an automorphism $\phi$ such that
$girth(G)<\infty$ whereas $girth(\mathbb{Z}\ltimes_{\phi}G)=\infty$. Let us
also mention (a somewhat relevant fact) that if $G$ is a finitely generated
group satisfying no law in two variables with $girth(G)<\infty$ then
$girth(((G\wr\mathbb{Z})\wr\mathbb{Z}))=\infty$ as shown in [A1].

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