compact generating set: if $X,Y$ are two, then in $G$ we have $Y\subset
X^{n}=X\cdots X$, and $X\subset Y^{n}$ for some $n$ by a Baire category
argument (some $Y^{l}$ must contain an open subset of $X$; it then follows
that every point of $X$ has a neighborhood in some $Y^{m}$, and by
compactness, $X$ is contained in some $Y^{n}$.) The relations on $X$ assuring
this involve words of length at most $n^{2}$; and any further relation on $Y$
of length $m$ translates to a relation on $X$ of length $mn$.

∎

######  Lemma C.3.

Let $A$ be a basic abelian group, and $H$ a group of automorphisms of $A$.
Assume each $H$-orbit in $A$ is precompact. Then a finite index subgroup
$H_{1}$ of $H$ acts trivially on $A/A^{0}$. Fixing such an $H_{1}$, one can
write $A$ as a direct sum of $H$-invariant subgroups, $A=C\oplus B$, where
$C\cong{\mathbb{R}}^{n}$, $H$ preserves an inner product on $C$,
$B\cong{\mathbb{Z}}^{k}$, and $H_{1}$ acts trivially on $B$.

###### Proof.

Let $C=A^{0}$; so $C\cong{\mathbb{R}}^{n}$. We first consider the conjugation
action of $H$ on $C$. Let $c_{1},\ldots,c_{n}\in C$ form a basis. Then
$c_{1}^{H},\cdots,c_{n}^{H}$ are precompact. Let $G$ be the subgroup of
$GL_{n}({\mathbb{R}})=End(C)$ preserving $cl(c_{i}^{H})$ for each $i$. Then it
is clear that $G$ is a compact subgroup of $GL_{n}({\mathbb{R}})$; hence
preserves an inner product on ${\mathbb{R}}^{n}$ (the integral over $G$ of the
$G$-conjugates of the standard inner product). Changing the isomorphism
$C\to{\mathbb{R}}^{n}$ we have $G\leq O_{n}$.

Next consider the action on $A/C\cong{\mathbb{Z}}^{k}$. Again let
$e_{1},\ldots,e_{k}$ be a basis. Then the orbit $E_{i}$ of $e_{i}$ is finite;
we have restriction maps $H\to Sym(E_{i})$; the common kernel of these maps is
a finite index subgroup $H_{1}$ of $H$, fixing each $e_{i}$ and hence fixing
$A/C$.

Claim .  For any $e\in A$, some point of $e+C$ is fixed by $H_{1}$.

###### Proof.

Fix $e$. Since $H_{1}$ fixes $A/C$, it leaves invariant the $C$-coset $e+C$.
Also, $H_{1}$-orbits are precompact; by taking the closed convex hull of at
most $n$ of them, we find a compact, convex, $H_{1}$-invariant set $B\subset
e+C$ left invariant by $H$. Since the Euclidean distance is preserved, so is
the measure and the center of mass of this convex set; it is a fixed point for
the action of $H_{1}$. ∎

Let $e_{1},\ldots,e_{k}$ be elements of $A$ generating $A$ modulo $C$.
Replacing $e_{i}$ by the $H$-fixed point of $e_{i}+C$, we may assume $e_{i}$
is fixed. Let $B=\langle e_{1},\ldots,e_{k}\rangle$. Then the decomposition we
promised is achieved.

∎

######  Lemma C.4.

[[61]] Let $G$ be a locally compact group. Assume $G/G^{0}$ is compact. Let
$B(G)$ be the union of all precompact conjugacy classes of $G$. Then $B(G)$ is
a closed subgroup of $G$.

###### Proof.

By Yamabe’s theorem, there exists an open subgroup $H$ of $G$ and an
$H$-normal compact subgroup $C_{1}$ of $1$, such that $H/C_{1}$ is a Lie
group. Since $G/H$ is discrete and $G^{0}$ is connected, we have $G^{0}\leq
H$. Now $G/G^{0}$ is compact, so $G/H$ is compact and discrete, hence finite.
Being normal in $H$, the group $C_{1}$ has only finitely many $G$-conjugates
$C_{1},\ldots,C_{r}$ in $G$. Let $C=\cap_{i=1}^{r}C_{i}$. Since $G/C_{i}$ is a
Lie group $L_{i}$, $G$ embeds into $\Pi_{i}L_{i}$; the map $G/C_{1}\to L_{1}$
is already proper, hence the same holds for $G/C\to\Pi_{i}L_{i}$, and it
follows that this map is closed; by Cartan’s theorem, $G/C$ is a Lie group.
Now $B(G/C)$ is closed by [61], Theorem 1, and hence $B(G)$ is closed by [61],
Proposition 1. ∎

######  Lemma C.5.

[ [22] Theorem 3.11]. Let $H$ be a locally compact topological group. Let $E$
be a conjugation-invariant subset of $H$ with compact closure and assume each
element $e\in E$ lies in a compact subgroup of $H$. Then $E$ is contained in a
compact subgroup of $H$.

######  Lemma C.6.

[[22] Lemma 3.16]. Let $G$ be a locally compact group. Assume every conjugacy
class in $G$ is precompact. Let $P$ be the union of all compact normal
subgroups of $G$. Then $P$ is a closed subgroup of $G$; and $G/P$ is the
product of a vector group with a discrete torsion-free abelian group.

######  Lemma C.7.

Let $H$ be a locally compact topological group. Assume $H$ has a compact open
subgroup $H_{0}$; notably this is the case if $H$ is totally disconnected. Let
$K$ be a normal precompact subset of $H$. Then there exists a compact normal
subgroup $N$ of $H$ such that the image $K/N$ of $K$ in $H/N$ is finite.

###### Proof.

$K^{\prime}:=K\cdot K^{-1}$ is also a precompact normal subset of $H$. Let
$E=\cup_{h\in H}h^{-1}H_{0}h\cap K^{\prime}$. Then $E$ is normal, precompact,
and each element $e\in E$ lies in a compact subgroup of $H$ (namely a
conjugate of $H_{0}$.) Let $N$ be the closure of the subgroup generated by
$E$. Then $N$ is normal, and by Lemma C.5, $N$ is compact. The image of $K$ in
$H/H_{0}$ is compact, while $H/H_{0}$ is discrete; hence it is finite. So $K$
is divided into finitely many cosets of $H_{0}$. But if $a,b\in K$ lie in the
same $H_{0}$-coset, then $ab^{-1}\in E\subset N$. Thus the image $K/N$ of $K$
modulo $N$ is finite.

∎

##  Appendix D On precompact conjugacy classes in locally compact groups

Groups whose every conjugacy class is finite (called FC-groups) were studied
by I. Schur, R. Baer, and B.H. Neumann, see [51]. This was later extended to
locally compact groups by Ushakov and (independently, it seems) Tits [61],
with emphasis on the connected case, and by [22] in general. Roughly speaking,
such groups are extensions of abelian groups by compact ones.

We will need a similar structure theorem (Theorem D.1) for a single precompact
conjugacy class. It seems likely that this exists in the literature, but I was
unable to find it. The result does not immediately reduce to the case where
all conjugacy classes are precompact, since as Tits showed, the union of all
compact conjugacy classes need not form a closed subgroup. Moreover, we need
to obtain commutativity not modulo an arbitrary compact normal subgroup, but
rather one generated by elements of the conjugacy class at hand. It is for the
latter purpose that we need Lemma C.2 and Lemma C.6. The reader interested
only in the structure theorem 5.11 for approximate subgroups, or the
applications to approximate lattices, can ignore this point; but it is
essential for the model-theoretic picture (e.g. approximate continuity, the
place of $G/G^{00}$ in ${\mathcal{G}}$, distinction between $X$ and
${\mathsf{S}}$) and probably for other group theoretic statements.

######  Theorem D.1.

Let $H$ be a locally compact topological group. Let $K$ be a normal precompact
subset of $H$. Then there exist closed normal subgroups $N\leq A\leq H_{1}\leq
H$ with $H/H_{1}$ finite, $N$ compact, with $A/N$ basic abelian and rigid in
$H_{1}/N$, such that $\langle K\rangle\cap H_{1}\leq A$. Moreover, $N,A\leq
cl(\langle K\rangle)$.

###### Proof.

Replacing $K$ by $cl(K\cup K^{-1})$, we may assume $K$ is symmetric and
compact. Let $H^{0}$ be the connected component of the identity in $H$.

Claim 1.  $H$ has a closed normal subgroup $M$ containing $H^{0}$, and a
finite index normal subgroup $H_{0}$ containing $M$, such that:

  1. (1)

$[K,H_{0}]\subset M$

  2. (2)

$M/H^{0}$ is compact

  3. (3)

$K/M$ is finite.

  4. (4)

$\langle K\rangle\cap H_{0}$ is generated by a precompact set $K_{0}$;
$K_{0}/M$ is finite.

  5. (5)

Let $D=cl\langle K_{0}M\rangle$ be the closed subgroup generated by $K_{0}M$.
Then $D/M\cong{\mathbb{Z}}^{k_{0}}$ for some ${k_{0}}\in{\mathbb{N}}$. It is
generated by some $a_{1},\ldots,a_{k_{0}}\in K_{0}$.

###### Proof.

By Lemma C.7 applied to $H/H^{0}$, there exists a closed normal subgroup $M$
of $H$ containing $H^{0}$, such that $M/H^{0}$ is compact, and the image of
$K$ in $H/M$ is finite. Let $H_{0}$ be the centralizer mod $M$ of the finite
normal set $K/M$ in $H/M$, i.e.

$H_{0}=\\{h\in H:(\forall k\in K)(hkM=khM)\\}$

So $H_{0}$ is closed and normal in $H$, of finite index; and (1,2,3) of Claim
1 hold.

Let $K_{0}=K^{\rho}\cap H_{0}$, where $\rho=[H:H_{0}]$. So $K_{0}$ is compact,
and $K_{0}/M$ is finite. Also $[K_{0},K_{0}]\subset M$

Claim 1a.  We have $\left\langle{K_{0}}\right\rangle=\langle K\rangle\cap
H_{0}$.

###### Proof.

Let $a_{1},\ldots,a_{n}$ be elements of $K$, whose product $a_{1}\cdots a_{n}$
lies in $H_{0}$. We have to show that $a_{1}\cdots a_{n}\in\langle
K_{0}\rangle$; we use induction on $n$. The case $n\leq\rho$ is trivial. If
$n>\rho$, let $b_{i}=a_{1}\cdots a_{i}$; then $b_{i}H_{0}=b_{j}H_{0}$ for some
$i<j\leq\rho$. Let $a=a_{1}\cdots a_{i}$, $b=a_{i+1}\cdots a_{j}$,
$c=a_{j+1}\cdots a_{n}$. Then $b\in K_{0}\cap H_{0}$; using normality,
$aba^{-1}\in K_{0}\subset\langle K_{0}\rangle$. By induction, $ac\in\langle
K_{0}\rangle$. Thus $a_{1}\cdots a_{n}=abc=(aba^{-1})ac\in\langle
K_{0}\rangle$. ∎

Thus (4) holds too. Now $D/M$ is generated by $K_{0}/M$ as a closed subgroup
of $H_{0}/M$, hence it is a locally compact abelian group. It is totally
disconnected since $H^{0}\leq M$. Thus $D/M$ has a profinite open subgroup
$D^{\prime}/M$; it is centralized by $H_{0}/M$, by (1). We may replace
$D^{\prime}$ by the join of the finitely many $H$-conjugates that it has, all
within the abelian group $D/M$; so we may take $D^{\prime}$ to be $H$-normal.
$D/D^{\prime}$ is a discrete abelian group, generated by the finite set
$K_{0}/D^{\prime}$. The torsion subgroup $D^{\prime\prime}/D^{\prime}$ of
$D/D^{\prime}$ is thus finite. Now $D^{\prime\prime}/D^{\prime},D^{\prime}/M$
and $M/H^{0}$ are compact, so $D^{\prime\prime}/H^{0}$ is compact. We may
replace $M$ by the bigger compact group $D^{\prime\prime}$, with no harm to
(1-4). Doing this, we find that ${cl\langle K_{0}M\rangle}/M$ is discrete,
torsion-free and finitely generated, hence $\cong{\mathbb{Z}}^{k_{0}}$ for
some ${k_{0}}$. This proves Claim 1. ∎

Since $D/M$ is discrete, it is generated algebraically by $K_{0}$, so
$cl\langle K_{0}\rangle M=\langle a_{1},\ldots,a_{k_{0}}\rangle M$ for some
$a_{1},\ldots,a_{k_{0}}\in K_{0}$.

Claim 2.  ${\langle K_{0}\rangle}$ is generated by $a_{1},\ldots,a_{k_{0}}$
along with $K_{1}:={K_{0}}^{r}\cap M$, provided $r$ is chosen large enough. We
have $\langle K_{1}\rangle=\langle K_{0}\rangle\cap M$, and so ${\langle
K_{0}\rangle}/\langle K_{1}\rangle$ is the free abelian group on
$a_{1},\ldots,a_{k_{0}}$.

###### Proof.

The group ${\mathbb{Z}}^{k_{0}}$ is finitely presented; hence for any
presentation in the category of locally compact groups, the relations are
compactly generated. The exact sequence $M\to cl\langle
K_{0}\rangle\to{\mathbb{Z}}^{k_{0}}$ gives such a presentation, with compact
generating set $K_{0}$. By Lemma C.2, the relations involving words of length
$\leq r$ generate all relations; this is precisely the statement of the claim.
∎

Since $H^{0}$ is connected, it is contained in (actually equal to) $M^{0}$. So
$M/M^{0}$ is compact. By Lemma C.4, the group $B(M)$ of elements of $M$ with
precompact conjugacy class is a closed normal subgroup of $M$. Note
$K_{0}^{r}\cap M\subset{K}^{r}\cap M\subset B(M)$. Also, by Claim 1 (1),
$[K,H_{0}]\subset M$; each element of $[K,H_{0}]$ has the form
$(aka^{-1})k^{-1}\in K^{2}$, so it has precompact conjugacy class; so
$[K,H_{0}]\subset B(M)$.

Let $P$ be the union of all compact normal subgroups of $B(M)$. Both $B(M)$
and $P$ are clearly normal in $H$. According to Lemma C.6, $P$ is a closed
subgroup of $B(M)$; and $B(M)/P$ is abelian. So $[K_{1},K_{1}]\subset P$. By
Lemma C.5, $cl\langle[K_{1},K_{1}]\rangle$ is a compact group; it is clearly
normal in $H$. We may factor out this compact normal subgroup, without
prejudice to the conclusion of the proposition (or of Claim 1.) Thus we may
assume that $[K_{1},K_{1}]=1$, i.e. $\langle K_{1}\rangle$ is abelian; and
hence $A_{1}:=cl\langle K_{1}\rangle$ is abelian.

By the structure of locally compact abelian groups ([48], Theorem 25), $A_{1}$
has an open subgroup $O_{1}$ isomorphic to $V\times{\mathbb{Z}}^{l}\times T$,
with $T$ a compact group and $V$ a vector group, i.e. $V\cong{\mathbb{R}}^{n}$
for some $n$. Since ${\mathbb{Z}}^{l}$ is discrete, there is in fact a smaller
open subgroup $O$ isomorphic to $V\times T$. Fix $O$, a vector group $V\leq
O$, and a compact $T\leq O$ with $O=V\oplus T$. Since $K_{1}$ is compact,
$K_{1}/O$ is finite; so $\langle K_{1}\rangle/O$ is finitely generated. Now
$\langle K_{1}\rangle$ is dense in $A_{1}$; so the image modulo $O$ is dense
in the discrete group $A_{1}/O$, i.e. equal to it. So $A_{1}/O$ is a finitely
generated Abelian group;
$A_{1}/O\cong{\mathbb{Z}}^{k_{1}}\oplus(O^{\prime}/O)$ where $O^{\prime}\geq
O$ and $O^{\prime}/O$ is the finite torsion subgroup of $A_{1}/O$. Let
$T^{\prime}$ be the divisible hull of $T$ within $O^{\prime}$; then
$T^{\prime}\cap O=T$ so $T^{\prime}/T$ is finite, and $T^{\prime}$ is compact.
Since $V$ is divisible, we have $O^{\prime}=V\oplus T^{\prime}$. Clearly,
$T^{\prime}$ is the unique maximal compact subgroup of $O^{\prime}$. If $C$ is
a compact subgroup of $A_{1}$, then $C/O^{\prime}$ is a finite group; but
$A_{1}/O^{\prime}$ is torsion-free, so $C\leq O^{\prime}$. Hence $T^{\prime}$
is unique maximal compact subgroup of $A_{1}$, so it is normal in $H$. We can
now further factor out by $T^{\prime}$. At this point we have: $V$ is open in
$A_{1}$. So $V=A_{1}^{0}$ is uniquely defined and normal in $H$ too.

Now $A_{1}/V=A_{1}/O^{\prime}\cong{\mathbb{Z}}^{k_{1}}$. Let
$a_{k_{0}+1},\ldots,a_{k}$ be elements of $A_{1}$, generating $A_{1}/V$. As we
saw above, we can in fact choose all the $a_{j}$ in $K_{1}$.

Let $A$ be the subgroup of $H_{0}$ generated by
$A_{1}\cup\\{a_{1},\ldots,a_{k_{0}}\\}$. By Claim 2, $K_{0}\subset\langle
K_{1},a_{1},\ldots,a_{k_{0}}\rangle\leq A$. Also, $A\cap M\subset<K_{0}>\cap
M\subset<K_{1}>\subset A_{1}$

Claim 3.  $A$ is closed.

###### Proof.

Let $c$ be a limit point of $A$; say $c=\lim_{i}c_{i},c_{i}\in A$ (sequence or
net). We have to prove that $c\in A$. Modulo $M$, $cM$ is a limit point of
$A/M$; but $A/M={\langle K_{0}\rangle}/M$ is closed and discrete (Claim 1(5)),
so $cM\in A/M$. Say $cM=aM,\ a\in A$; replacing $c$ by $ca^{-1}$ and $c_{i}$
by $c_{i}a^{-1}$, we may assume that $c\in M$. Similarly since $c_{i}\to c$,
we have $c_{i}M\to cM=M$, and by discreteness we must have $c_{i}\in M$ for
all sufficiently large $i$. Then $c_{i}\in A\cap M=A_{1}$. Since $A_{1}$ is
closed, $c\in A$. ∎

So $A$ is locally compact. Now $A_{1}/V$ is countable, and $A/A_{1}$ is
countable, hence $A/V$ is a countable locally compact group; this implies that
it is discrete. So $A$ is basic abelian. The $H$-orbit $a_{i}^{H}/V$ of each
$a_{i},i=1,\ldots,k$ in $A/V$ is thus finite (being compact; as each $a_{i}$
lies in $K$.) Let $H_{1}$ be the mod-$V$-centralizer in $H$ of these orbits:

$H_{1}:=\\{h\in H_{0}:\bigwedge_{i\leq n}(\forall c\in a_{i}^{H})\ \
hcV=chV\\}$

Then $H_{1}\leq H_{0}$, and $H_{1}$ has finite index in $H$. Thus the
connected group $V$ is a subgroup of $H_{1}$. We replace $A$ by $A\cap H_{1}$.
As in Claim 1a, $\langle K\rangle\cap H_{1}$ is generated by a compact set,
and $\langle K\rangle\cap H_{1}=(\langle K\rangle\cap H_{0})\cap H_{1}=\langle
K_{0}\rangle\cap H_{1}=\langle
K_{1}\cup\\{a_{1},\ldots,a_{k_{0}}\\}\rangle\cap H_{1}\leq A\cap H_{1}$.

By Lemma C.3, since $H_{1}$ acts trivially on $A/A^{0}$, $A$ is rigid in
$H_{1}$. ∎

## References

  * [1] Louis Auslander, On radicals of discrete subgroups of Lie groups, American Journal of Mathematics 85 (1963), no. 2, 145-150.
  * [2] Reinhold Baer, Finiteness properties of groups, Duke Math. J. 15 (1948), 1021-32.
  * [3] Itaï Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Transactions of the American Mathematical Society 362 (2010), no. 10, 5213–5259
  * [4] Michael Björklund and Tobias Hartnick, Approximate lattices. Duke Math. J., 167(15):2903-2964, 2018
  * [5] Michael Björklund, Tobias Hartnick, Thierry Stulemeijer, Borel density for approximate lattices, Forum of Mathematics, Sigma (2019), Vol. 7, e40
  * [6] M. Björklund and A. Fish, Approximate invariance for ergodic actions of amenable groups, preprint, arXiv:1607.02575v3 (2018)
  * [7] Breuillard, Emmanuel; Green, Ben; Tao, Terence, The structure of approximate groups. Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221
  * [8] A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72(1) (1960), 179-188.
  * [9] Armand Borel and Jacques Tits, Homomorphismes "Abstraits" de Groupes Algebriques Simples, Annals of Mathematics, Second Series, Vol. 97, No. 3 (May, 1973), pp. 499-571
  * [10] R. Brooks, Some remarks on bounded cohomology, In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), 53-63, Princeton, N.J., 1981. Princeton Univ. Press
  * [11] Burger, M.; Monod, N. Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12 (2002), no. 2, 21–280
  * [12] Caprace, Pierre-Emmanuel; Monod, Nicolas, Future directions in locally compact groups: a tentative problem list. In New directions in locally compact groups, pp. 343–355, London Math. Soc. Lecture Note Ser., 447, Cambridge Univ. Press, Cambridge, 2018.
  * [13] Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990.
  * [14] Enrique Casanovas, D. Lascar, Anand Pillay, M. Ziegler, Galois Groups of first-order theories, Journal of Mathematical Logic 01(02), January 2012 DOI: 10.1142/S0219061301000119
  * [15] Dixon, John D, Nilpotence and Local Nilpotence of Linear Groups, Linear algebra and its applications 13, 5-7 (1976)
  * [16] Ellis, Robert, Locally compact transformation groups. Duke Math. J. 24 (1957), 119-125.
  * [17] Arturo Rodríguez Fanlo, On piecewise hyperdefinable groups.
  * [18] Koji Fujiwara and Michael Kapovich, On quasihomomorphisms with noncommutative targets. Geom. Funct. Anal. Vol. 26 (2016) 478-519
  * [19] Harry Furstenberg, A Note on Borel’s Density Theorem, Proceedings of the American Mathematical Society , Feb., 1976, Vol. 55, No. 1 (Feb., 1976), pp. 209-212
  * [20] Green, Ben; Sanders, Tom, A quantitative version of the idempotent theorem in harmonic analysis, Annals of Mathematics, 168 (2008), 1025–1054.
  * [21] Gromov,M., Volume and bounded cohomology. Inst. Hautes Études Sci. Publ.Math.(56) (1982), 5–99
  * [22] S. Grosser, M. Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971) 1-40.
  * [23] Helfgott, H. A. Growth and generation in ${\rm SL}_{2}(\mathbb{Z}/p\mathbb{Z})$. Ann. of Math. (2) 167 (2008), no. 2, 601–623.
  * [24] Ehud Hrushovski., Stability and approximate subgroups.
  * [25] Ehud Hrushovski, Definability patterns and their symmetries, ArXiv. ‘
  * [26] Ehud Hrushovski, Krzysztof Krupiński, Anand Pillay, Amenability and definability, arXiv:1901.02859
  * [27] Hrushovski, Ehud; Peterzil, Ya’acov; Pillay, Anand, Groups, measures, and the NIP. J. Amer. Math. Soc. 21 (2008), no. 2, 563–596.
  * [28] Ivanov, N. V. Foundations of the theory of bounded cohomology. (Russian) Studies in topology, V. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143 (1985), 69–109, 177–178. English translation in Journal of Soviet Mathematics volume 37, pages1090–1115 (1987)
  * [29] Kaplan, Itay; Miller, Benjamin D; Simon, Pierre: The Borel cardinality of Lascar strong types. J. Lond. Math. Soc. (2), 90 (2), pp. 609–630, 2014
  * [30] Kaplan, Itay; Ramsey, Nicholas, On Kim-independence. J. Eur. Math. Soc. (JEMS) 22 (2020), no. 5, 1423–1474.)
  * [31] Kazhdan, D. On $\epsilon$-representations. Israel J. Math. 43 (1982), no. 4, 315–323
  * [32] Kim, Byunghan; Pillay, Anand Simple theories. Joint AILA-KGS Model Theory Meeting (Florence, 1995). Ann. Pure Appl. Logic 88 (1997), no. 2-3, 149–164.
  * [33] Krupiński, Krzysztof; Pillay, Anand, Generalised Bohr compactification and model-theoretic connected components. Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 2, 219–249
  * [34] Krupiński, Krzysztof; Pillay, Anand; Rzepecki, Tomasz, Topological dynamics and the complexity of strong types. Israel J. Math. 228 (2018), no. 2, 863-932.
  * [35] K Krupiński, L Newelski, P Simon, Boundedness and absoluteness of some dynamical invariants in model theory, arXiv:1705.00159, 2017.
  * [36] A. S. Kechris, S. Solecki and S. Todorcevic, Borel Chromatic Numbers, Advances in Mathematics 141, 144 (1999).
  * [37] D. Lascar, On the category of models of a complete theory, J. Symbolic Logic 47 (1982), no. 2, 249–266
  * [38] D. Kotschick, What is a quasi-morphism? Notices of the AMS v. 51, no. 2, February 2004, pp. 208-209
  * [39] Simon Machado, Approximate Lattices and Meyer Sets in Nilpotent Lie Groups, Discrete analysis 2020:1
  * [40] Simon Machado, Infinite approximate subgroups of soluble lie groups, arXiv:1909.12151v1
  * [41] Simon Machado, Good Models, Infinite Approximate Subgroups and Approximate Lattices, arXiv:2011.01829 math.GR
  * [42] Simon Machado, Approximate lattices in higher-rank semi-simple groups, arXiv:2011.01835 [math.GR]
  * [43] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups. Springer, Berlin Heidelberg New York, 1991. ISBN 3-540-12179-X,
  * [44] Jean-Cyrille Massicot & Frank O. Wagner, Approximate subgroups, Journal de l’‘Ecole polytechnique Tome 2 (2015), p. 55-63. http://jep.cedram.org/item?id=JEP_2015__2__55_0
  * [45] Yves Meyer, Algebraic numbers and harmonic analysis, volume 2. Elsevier, 1972.
  * [46] Monod, Nicolas, On the bounded cohomology of semi-simple groups, S-arithmetic groups and products, J. Reine Angew. Math. 640 (2010), 167–202
  * [47] Monod, Nicolas, An invitation to bounded cohomology. International Congress of Mathematicians. Vol. II, 1183-1211, Eur. Math. Soc., Zürich, 2006.
  * [48] Sidney A. Morris, Pontryagin duality and the structure of locally compact abelian groups, London Math, Soc, Lecture Note Series No. 29, Cambridge Univ, Press, 1977
  * [49] Morris, Dave Witte, Introduction to arithmetic groups. Deductive Press, 2015. xii+475 pp
  * [50] Mostow, G. D. Arithmetic subgroups of groups with radical. Ann. of Math. (2) 93 (1971), 409-438.
  * [51] B. H. Neumann, Groups with Finite Classes of Conjugate Elements (In Memoriam Issai Schur), Proc. LMS (3) 1 (1951)
  * [52] Hee Oh, Adelic version of Margulis arithmeticity theorem. Math. Ann. 321 (2001), no. 4, 789-815.
  * [53] Noskov, G. A. Bounded cohomology of discrete groups with coefficients. (Russian) Algebra i Analiz 2 (1990), no. 5, 146–164; translation in Leningrad Math. J. 2 (1991), no. 5, 1067–1084
  * [54] Gopal Prasad, Elementary proof of a theorem of Bruhat-Tits- Rousseau and of a theorem of Tits, Bulletin de la S. M. F., tome 110 (1982), p. 197–202
  * [55] T. Sanders, On a non-abelian Balog-Szemeredi-type lemma, J. Aust. Math. Soc. 89 (2010), 127-132.
  * [56] Shelah, Saharon, A two-cardinal theorem. Proc. Amer. Math. Soc. 48 (1975), 207-213.
  * [57] Shelah, S. Classification theory and the number of nonisomorphic models. Studies in Logic and the Foundations of Mathematics, 92. North-Holland Publishing Co., Amsterdam, 1978, Second edition.1990.
  * [58] Shelah, Saharon, Simple unstable theories. Ann. Math. Logic 19 (1980), no. 3, 177–203.
  * [59] S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 (1996) 229-255.
  * [60] Tao, Terence, Product set estimates for non-commutative groups. Combinatorica 28 (2008), no. 5, 547-594.
  * [61] Tits, J., Automorphismes à déplacement borné des groupes de Lie. Topology 3 (1964), no. suppl, suppl. 1, 97-107
  * [62] Turing, A. M. Finite approximations to Lie groups. Ann. of Math. (2) 39 (1938), no. 1, 105–111
  * [63] Yamabe, Hidehiko A generalization of a theorem of Gleason. Ann. of Math. (2) 58, (1953). 351–365.
  * [64] Wang, S. P., On density properties of S-subgroups of locally compact groups. Ann. of Math. (2) 94 (1971), 325-329.


$\displaystyle\left|\int_{\pi/2}^{\theta_{i}}\frac{\cos^{2}\theta\text{d$\theta$}}{\sqrt{\hat{\Theta}(\theta)}}\right|=\dfrac{1}{\sqrt{-u_{-}a^{2}}}\int_{0}^{\hat{x}_{i}}\dfrac{\hat{u}_{+}\sin^{2}x\text{dx}}{\sqrt{1-k\sin^{2}x}}=\dfrac{1}{\sqrt{-u_{-}a^{2}}}\frac{\hat{u}_{+}(F(\hat{x}_{i}|\hat{k})-E(\hat{x}_{i}|\hat{k}))}{\hat{k}}.$
(171)

Moreover, in order to obtain the time delay and azimuthal angle parameter, we
employ a similar approach as in the case of kds. This involves integrating
over half orbits and treating the radial coordinate as a constant. Thus, we
express eq. 164 and (165) as,

$\displaystyle\hat{\delta}=\left(\frac{ar^{2}+a^{3}-a\hat{\Delta}-a^{2}\hat{\lambda}}{\hat{\Delta}}\right)\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\text{d$\theta$}}{\sqrt{\hat{\Theta}(\theta)}}\right|+\hat{\lambda}\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\text{d$\theta$}}{\sin^{2}\theta\sqrt{\hat{\Theta}(\theta)}}\right|,$
(172)
$\displaystyle\hat{\zeta}=\left(\frac{(r^{2}+a^{2})(r^{2}+a^{2}-a\lambda)}{\hat{\Delta}}+a\hat{\lambda}-a^{2}\right)\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\text{d$\theta$}}{\sqrt{\hat{\Theta}(\theta)}}\right|+a^{2}\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\cos^{2}\theta\text{d$\theta$}}{\sqrt{\hat{\Theta}(\theta)}}\right|,$
(173)

where we have employed the equality relation given by eq. 125 to transform the
Mino parameter into an angular integral. Making use of the general solutions
eq. 168-(171), we obtain,

$\displaystyle\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\text{d$\theta$}}{\sqrt{\hat{\Theta}(\theta)}}\right|=\dfrac{2}{\sqrt{-\hat{u}_{-}a^{2}}}K\left(\hat{k}\right)=\hat{\Gamma}_{\theta},\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\cos^{2}\theta\text{d$\theta$}}{\sqrt{\hat{\Theta}(\theta)}}\right|=\dfrac{2\hat{u}_{+}}{\hat{k}\sqrt{-u_{-}a^{2}}}\left(K(\hat{k})-E(\hat{k})\right)=\hat{\Gamma}_{t},$
(174)
$\displaystyle\left|\int_{\theta_{1}}^{\theta_{4}}\frac{\text{d$\theta$}}{\sin^{2}\theta\sqrt{\hat{\Theta}(\theta)}}\right|=\dfrac{2}{\sqrt{-u_{-}a^{2}}}\Pi(\hat{u}_{+}|\hat{k})=\hat{\Gamma}_{\phi}.$
(175)

Moreover, the Lyapunov exponent per half orbits is derived using the same
approach in kds through the relation,

$\displaystyle\hat{\gamma}=\sqrt{\frac{\hat{R^{\prime\prime}}(r_{b})}{2}}\hat{\Gamma}_{\theta}.$
(176)

To obtain solutions for the radial integrals, firstly we express the radial
potential and the parameter $\hat{\delta}_{r}$ in terms of their roots,

$\displaystyle\hat{R}(r)=(r-\hat{r}_{1})(r-\hat{r}_{2})(r-\hat{r}_{3})(r-\hat{r}_{4}),\hat{\Delta}=-\frac{3}{\Lambda}(r-\hat{r}_{-c})(r-\hat{r}_{-})(r-\hat{r}_{+})(r-\hat{r}_{c}).$
(177)

Using eq. 177 we rewrite eq. 163-(165) as,

$\displaystyle\hat{I}_{r}=\int_{r_{s}}^{r_{o}}\frac{\text{dr}}{\pm\sqrt{(r-\hat{r}_{1})(r-\hat{r}_{2})(r-\hat{r}_{3})(r-\hat{r}_{4})}},$
(178)
$\displaystyle\hat{\phi}_{o}-\hat{\phi}_{s}=a\left(\frac{\hat{\Omega}_{-c}}{r_{c,-c}r_{-,-c}r_{+,-c}}\hat{I}_{-c}+\frac{\hat{\Omega}_{-}}{r_{c,-}r_{-c,-}r_{+,-}}\hat{I}_{-}+\frac{\hat{\Omega}_{+}}{r_{c,+}r_{-c,+}r_{-,+}}\hat{I}_{+}+\frac{\hat{\Omega}_{c}}{r_{-c,c}r_{-,c}r_{+,c}}\hat{I}_{c}-\tau\right)+\hat{\lambda}\hat{I}_{\phi},$
(179)
$\displaystyle\hat{t}_{o}-\hat{t}_{s}=\frac{\hat{\Phi}_{c}}{\hat{r}_{-c,c}\hat{r}_{-,c}\hat{r}_{+,c}}+\frac{\hat{\Phi}_{-c}}{\hat{r}_{c,-c}\hat{r}_{c,-c}\hat{r}_{+,-c}}+\frac{\hat{\Phi}_{-}}{\hat{r}_{c,-}\hat{r}_{-c,-}\hat{r}_{+,-}}+\frac{\hat{\Phi}_{+}}{\hat{r}_{c,+}\hat{r}_{-c,+}\hat{r}_{-,+}}+\tau\left(-a^{2}+a\hat{\lambda}-\frac{\Lambda}{3}\right),$
(180)
$\displaystyle\hat{\Phi}_{x}=\left(a^{2}+\hat{r}_{x}^{2}\right)\hat{\Omega}_{x}\hat{I}_{x},\hat{\Omega}_{x}=\frac{3}{\Lambda}\left(a^{2}-a\hat{\lambda}+\hat{r}_{x}^{2}\right),\hat{I}_{x}=\int_{r_{s}}^{r_{o}}\frac{\text{dr}}{(r-\hat{r}_{x})\sqrt{\hat{R}(r)}}.$
(181)

The subscripts $x$ denotes the horizons $-c,-,+,c$, where $\hat{r}_{+}$,
$\hat{r}_{-}$, $\hat{r}_{c}$, and $\hat{r}_{-c}$ represent the event horizon,
Cauchy horizon, the cosmological horizon and its dual, respectively. Further,
$\hat{r}_{a,b}=\hat{r}_{a}-r_{b}$. Further, the subscript $o$ and $s$ denote
the source and the observer respectively, where $r_{o}$ and $r_{s}$ need to be
chosen within the domain of outer communication. The radial integrals
$\hat{I}_{r}$ and $\hat{I}_{x}$ takes the same form as that of kds. In the
same way we obtain their solutions for the roots
($\hat{r}_{1}<\hat{r}_{2}<\hat{r}_{3}<\hat{r}_{4}<r_{s}$) and
($\hat{r}_{1}<\hat{r}_{2}<r_{s},\hat{r}_{3}=\bar{\hat{r}}_{4}$).

  * •

$\hat{r}_{1}<\hat{r}_{2}<\hat{r}_{3}<\hat{r}_{4}<r_{s}$

$\displaystyle\int_{r_{4}}^{r_{i}}\frac{\text{dr}}{\pm\sqrt{(r-\hat{r}_{1})(r-\hat{r}_{2})(r-\hat{r}_{3})(r-\hat{r}_{4})}}=\hat{g}_{E}F(\hat{\varphi}_{E,i}|\hat{k}_{E}),$
(182)
$\displaystyle\int_{r_{4}}^{r_{i}}\frac{\text{dr}}{{\pm_{r}}(r-\hat{r}_{x})\sqrt{(r-\hat{r}_{1})(r-\hat{r}_{2})(r-\hat{r}_{3})(r-\hat{r}_{4})}}=-\frac{2}{\hat{r}_{x,3}.\sqrt{\hat{r}_{4,2}\hat{r}_{3,1}}}\left(F\left(\hat{\varphi}_{E,i}|\hat{k}_{E}\right)+\frac{\hat{r}_{4,3}\hat{\Pi}_{i})}{\hat{r}_{x,4}}\right),$
(183)

where we have defined,

$\displaystyle\hat{\varphi}_{E,i}=\arcsin\sqrt{\frac{\hat{r}_{3,1}\hat{r}_{i,4}}{\hat{r}_{4,1}\hat{r}_{i,3}}},\hat{k}_{E}=\frac{\hat{r}_{4,1}\hat{r}_{3,2}}{\hat{r}_{4,2}\hat{r}_{3,1}},\hat{g}_{E}=\frac{2}{\sqrt{\hat{r}_{4,2}\hat{r}_{3,1}}},\hat{\Pi}_{i}=\Pi\left(\frac{\hat{r}_{4,1}\hat{r}_{x,3}}{\hat{r}_{x,4}\hat{r}_{3,1}};\hat{\varphi}_{E,i}|\hat{k}_{E}\right)$
(184)

  * •

$\hat{r}_{1}<\hat{r}_{2}<r_{s},\hat{r}_{3}=\bar{\hat{r}}_{4}$
In the same way as eq. 155 and (156) we obtain the solution for this case in
rkds as,

$\displaystyle\int_{r_{2}}^{r_{i}}\frac{\text{dr}}{\pm\sqrt{(r-\hat{r}_{1})(r-\hat{r}_{2})(r-\hat{r}_{3})(r-\hat{r}_{4})}}=\hat{g}_{P}F(\hat{\varphi}_{P,i}\mid
k_{P}),$ (185)
$\displaystyle\int_{r_{2}}^{r_{i}}\frac{\text{dr}}{{\pm_{r}}(r-\hat{r}_{x})\sqrt{(r-\hat{r}_{1})(r-\hat{r}_{2})(r-\hat{r}_{3})(r-\hat{r}_{4})}}=\frac{(\hat{B}-\hat{A})\left((\hat{\alpha}-\hat{\alpha}_{2})\hat{R}_{1,i}+\hat{\alpha}_{2}F(\hat{\varphi}_{P,i}\mid\hat{k}_{P})\right)}{\sqrt{\hat{A}\hat{B}}(\hat{r}_{2}\hat{B}+\hat{r}_{1}\hat{A}-\hat{A}\hat{r}_{x}-\hat{B}\hat{r}_{x})}.$
(186)

The parameters $\hat{A},\hat{B},\hat{k}_{P}$ have been defined as,

$\displaystyle\hat{A}=\sqrt{\left(\hat{r}_{2}-\frac{\hat{r}_{3}+\hat{r}_{4}}{2}\right)^{2}-\frac{(\hat{r}_{3}-\hat{r}_{4})^{2}}{4}},\hat{B}=\sqrt{\left(\hat{r}_{1}-\frac{\hat{r}_{3}+\hat{r}_{4}}{2}\right)^{2}-\frac{(\hat{r}_{3}-\hat{r}_{4})^{2}}{4}},$
(187)
$\displaystyle\hat{k}_{P}=\frac{(\hat{A}+\hat{B})^{2}-(\hat{r}_{2}-\hat{r}_{1})^{2}}{4\hat{A}\hat{B}}.$
(188)

We have also defined,

$\displaystyle\hat{R}_{1,i}=\frac{1}{1-\hat{\alpha}^{2}}\left(\Pi\left[\frac{\hat{\alpha}^{2}}{\hat{\alpha}^{2}-1};\hat{\varphi}_{P,i}\mid\hat{k}_{P}\right]-\hat{\alpha}\hat{f}_{1,i}\right),\quad\hat{\alpha}=\frac{\hat{B}r_{2}+\hat{A}r_{1}-r_{x}(A+\hat{B})}{\hat{B}r_{2}-r_{1}\hat{A}+r_{x}(\hat{A}+\hat{B})},\hat{\alpha}_{2}=\frac{\hat{A}+\hat{B}}{\hat{B}-\hat{A}},$
(189)
$\displaystyle\hat{f}_{1,i}=\sqrt{\frac{\hat{\alpha}^{2}-1}{\hat{k}_{P}+(1-\hat{k}_{P})\hat{\alpha}^{2}}}\ln{\left|\frac{\sqrt{\hat{k}_{P}+(1-\hat{k}_{P})\hat{\alpha}^{2}}dn(\hat{u}_{i}\mid\hat{k}_{P})+\sqrt{\hat{\alpha}^{2}-1}sn(\hat{u}_{i}\mid\hat{k}_{P})}{\sqrt{\hat{k}_{P}+(1-\hat{k}_{P})\hat{\alpha}^{2}}dn(\hat{u}_{i}\mid\hat{k}_{P})-\sqrt{\hat{\alpha}^{2}-1}sn(\hat{u}_{i}\mid\hat{k}_{P})}\right|},$
(190)
$\displaystyle\hat{u}_{i}=cn^{-1}\left[\cos(\hat{\varphi}_{P,i})\mid\hat{k}_{P}\right],\hat{\varphi}_{P,i}=\arccos\left[\frac{(\hat{A}-\hat{B})r_{i}+\hat{r}_{2}\hat{B}-\hat{r}_{1}A}{(\hat{A}+\hat{B})r_{i}-\hat{r}_{2}\hat{B}-\hat{r}_{1}\hat{A}}\right].$
(191)

##  Appendix D Integrals in terms of angular turning points

For purposes of analytic ray tracing, it becomes imperative to express the
solutions in terms of the turning points experienced by photons. In this
section we will express the angular integrals in terms of the number of
turning points that the photons encounter in the $\theta$ direction.
Considering all possible configurations, Ref. [31] demonstrates that angular
path integrals for ordinary motion unpack as,

$\displaystyle\int_{\theta_{s}}^{\theta_{o}}=2m\left|\int_{\pi/2}^{\theta_{\pm}}\right|+\eta_{s}\left|\int_{\pi/2}^{\theta_{s}}\right|-\eta_{o}\left|\int_{\pi/2}^{\theta_{o}}\right|.$
(193)

Where,

$\displaystyle\eta_{s}=sign(p^{\theta}_{s})sign(\cos\theta_{s})=(-1)^{m}sign(p^{\theta}_{o})sign(\cos\theta_{s}),\eta_{o}=sign(p^{\theta}_{o})sign(\cos\theta_{o})=(-1)^{m}sign(p^{\theta}_{s})sign(\cos\theta_{o}),$
(194)

and $m$ is the number of turning points in the latitudinal direction. We have
already obtained all the general solutions thus we will directly apply our
solutions to eq. 193.

In kds spacetime we obtain,

$\displaystyle\mathbb{I}_{\theta}=\dfrac{2m}{\sqrt{-u_{-}\Xi}}K\left(k\right)+\dfrac{(-1)^{m}sign(p^{\theta}_{o})}{\sqrt{-u_{-}\Xi}}F\left(x_{s}|k\right)-\dfrac{sign(p^{\theta}_{o})}{\sqrt{-u_{-}\Xi}}F\left(x_{o}|k\right),$
(195)
$\displaystyle\mathbb{I}_{\phi}=\dfrac{2m}{\sqrt{-u_{-}\Xi}}\Pi\left(-u_{+}\mathcal{Y};|k\right)+\dfrac{(-1)^{m}sign(p^{\theta}_{o})}{\sqrt{-u_{-}\Xi}}\Pi\left(-u_{+}\mathcal{Y};x_{s}|k\right)-\dfrac{sign(p^{\theta}_{o})}{\sqrt{-u_{-}\Xi}}\Pi\left(-u_{+}\mathcal{Y};x_{o}|k\right),$
(196)
$\displaystyle\mathbb{I}_{t}=\frac{2mu_{+}J(-u_{+}\mathcal{Y}|k)}{\sqrt{-u_{-}\Xi}}+\frac{(-1)^{m}sign(p^{\theta}_{o})u_{+}J(-u_{+}\mathcal{Y},\arcsin
t_{s}|k)}{\sqrt{-u_{-}\Xi}}-\frac{sign(p^{\theta}_{o})u_{+}J(-u_{+}\mathcal{Y},\arcsin
t_{o}|k)}{\sqrt{-u_{-}\Xi}},$ (197)
$\bar{\mathbb{I}}_{\phi}=\dfrac{2m}{\sqrt{-u_{-}\Xi}}\dfrac{1}{1+\mathcal{Y}}[\Pi(u_{+}|k)+\mathcal{Y}\Pi(-u_{+}\mathcal{Y}|k)]+\dfrac{(-1)^{m}sign(p^{\theta}_{o})}{\sqrt{-u_{-}\Xi}}\dfrac{1}{1+\mathcal{Y}}[\Pi(u_{+};\arcsin
t_{s}|k)+\mathcal{Y}\Pi(-u_{+}\mathcal{Y};\arcsin t_{s}|k)]\\\
-\dfrac{sign(p^{\theta}_{o})}{\sqrt{-u_{-}\Xi}}\dfrac{1}{1+\mathcal{Y}}[\Pi(u_{+};\arcsin
t_{o}|k)+\mathcal{Y}\Pi(-u_{+}\mathcal{Y};\arcsin t_{o}|k)],$ (198)

while in rkds spacetime we obtain,

$\displaystyle\hat{\mathbb{I}}_{\theta}=\dfrac{2m}{\sqrt{-\hat{u}_{-}a^{2}}}K\left(\hat{k}\right)+\dfrac{(-1)^{m}sign(p^{\theta}_{o})}{\sqrt{-\hat{u}_{-}a^{2}}}F\left(\hat{x}_{s}|\hat{k}\right)-\dfrac{sign(p^{\theta}_{o})}{\sqrt{-\hat{u}_{-}a^{2}}}F\left(\hat{x}_{o}|\hat{k}\right),$
(199)
$\displaystyle\hat{\mathbb{I}}_{\phi}=\dfrac{2m}{\sqrt{-\hat{u}a^{2}}}\Pi(\hat{u}_{+}|\hat{k})+\dfrac{(-1)^{m}sign(p^{\theta}_{o})}{\sqrt{-\hat{u}a^{2}}}\Pi[\hat{u}_{+};\hat{x}_{s}|\hat{k}]-\dfrac{sign(p^{\theta}_{o})}{\sqrt{-\hat{u}a^{2}}}\Pi[\hat{u}_{+};\hat{x}_{o}|\hat{k}],$
(200)
$\hat{\mathbb{I}}_{t}=\dfrac{2m}{\sqrt{-\hat{u}a^{2}}}\frac{\hat{u}_{+}(F(\hat{k})-E(\hat{k}))}{\hat{k}}+\dfrac{(-1)^{m}sign(p^{\theta}_{o})}{\sqrt{-\hat{u}a^{2}}}\frac{\hat{u}_{+}(F(\hat{x}_{s}|\hat{k})-E(\hat{x}_{s}|\hat{k}))}{\hat{k}}\\\
-\dfrac{sign(p^{\theta}_{o})}{\sqrt{-\hat{u}a^{2}}}\frac{\hat{u}_{+}(F(\hat{x}_{o}|\hat{k})-E(\hat{x}_{o}|\hat{k}))}{\hat{k}}.$
(201)

The $sign(p^{\theta}_{o})$ is equivalent to $sign(y)$. The above equations are
only valid for observers at latitude $\theta_{o}\in(0,\pi/2)$.
We postpone the case of $\theta_{o}=0,\pi/2$ to our future work.

##  Appendix E Change in $\phi$ and $t$ for subsequent images

In section III, an analysis of direct images, lensing rings and photon rings
has been done. In the same section we have slightly talked about subsequent
images, ($(m+1)^{th}$,$m^{th}$), undergoing a rotation due to change in
azimuthal angle and a delay in time of detection. In this part we explicitly
derive these quantities. Giving the explicit forms of these quantities is
important as we shall see that they are related to the critical parameters
controlling the photon ring structure.

To obtain the kds change in azimuthal angle between subsequent images
$\Delta\phi_{m+1}-\Delta\phi_{m}$ we proceed as follows,

$\Delta\phi_{m+1}-\Delta\phi_{m}=\left(\int_{r_{o}}^{r_{s}}\frac{\left(aL^{2}\right)\left(a(a-\lambda)+\text{r}^{2}\right)\text{dr}}{\pm_{r}\text{$\Delta_{r}$}\sqrt{R(r)}}-L^{2}a\mathbb{I}_{\phi}+L^{2}\lambda\bar{\mathbb{I}}_{\phi}\right)_{m+1}-\\\
\left(\int_{r_{s}}^{r_{o}}\frac{\left(aL^{2}\right)\left(a(a-\lambda)+\text{r}^{2}\right)\text{dr}}{\pm_{r}\text{$\Delta_{r}$}\sqrt{R(r)}}-L^{2}a\mathbb{I}_{\phi}+L^{2}\lambda\bar{\mathbb{I}}_{\phi}\right)_{m},$
(202a)
$=\int_{r_{s}^{m+1}}^{r_{s}^{m}}\frac{\left(aL^{2}\right)\left(a(a-\lambda)+\text{r}^{2}\right)\text{dr}}{\pm_{r}\text{$\Delta_{r}$}\sqrt{R(r)}}-\dfrac{2L^{2}a}{\sqrt{-u_{-}\Xi}}\Pi\left(-u_{+}\mathcal{Y};|k\right)+\dfrac{2L^{2}\lambda}{(1+\mathcal{Y})\sqrt{-u_{-}\Xi}}[\Pi(u_{+}|k)+\mathcal{Y}\Pi(-u_{+}\mathcal{Y}|k)],$
(202b)
$=\frac{\left(aL^{2}\right)\left(a(a-\lambda)+\text{r}^{2}\right)}{\text{$\Delta_{r}$}}\Gamma_{\theta}-\dfrac{2L^{2}a}{\sqrt{-u_{-}\Xi}}\Pi\left(-u_{+}\mathcal{Y};|k\right)+\dfrac{2L^{2}\lambda}{(1+\mathcal{Y})\sqrt{-u_{-}\Xi}}[\Pi(u_{+}|k)+\mathcal{Y}\Pi(-u_{+}\mathcal{Y}|k)].$
(202c)

In eq. 202b we have performed a subtraction alongside inserting eq. 196 and
eq. 198. Subsequent images are just images of same sources that have undergone
extreme lensing. Therefore, for each image, $r_{s}^{m+1}$ is simply the same
as $r_{s}^{m}$. Thus on the radial integral, we have kept $r$ to be constant.
Additionally, $r_{s}^{m}$ to $r_{s}^{m+1}$ is equivalent to half an orbit. The
Mino period for half an orbit is as given in eq. 137c. This is the approach we
have used to arrive at eq. 202c.

Performing the same analysis for $\Delta t$ gives us,

$\Delta t_{m+1}-\Delta
t_{m}=\dfrac{L^{2}((r^{2}+a^{2})^{2}-a\lambda(a^{2}+r^{2}))}{\Delta_{r}}\Gamma_{\theta}+\frac{2a^{2}L^{2}u_{+}J(-u_{+}\mathcal{Y}|k)}{\sqrt{-u_{-}\Xi}}-\dfrac{2aL^{2}(a-\lambda)}{\sqrt{-u_{-}\Xi}}\Pi\left(-u_{+}\mathcal{Y};|k\right).$
(203)

Likewise in rkds spacetime we obtain these parameters as,

$\displaystyle\hat{\Delta}\phi_{m+1}-\hat{\Delta}\phi_{m}=\left(\frac{ar^{2}+a^{3}-a\hat{\Delta}-a^{2}\hat{\lambda}}{\hat{\Delta}}\right)\hat{\Gamma}_{\theta}+\dfrac{2\hat{\lambda}}{\sqrt{-u_{-}a^{2}}}\Pi(\hat{u}_{+}|\hat{k}),$
(204)
$\displaystyle\hat{\Delta}t_{m+1}-\hat{\Delta}t_{m}=\left(\frac{(r^{2}+a^{2})(r^{2}+a^{2}-a\lambda)}{\hat{\Delta}}+a\hat{\lambda}-a^{2}\right)\hat{\Gamma}_{\theta}+\dfrac{2a^{2}}{\sqrt{-u_{-}a^{2}}}\frac{\hat{u}_{+}(E(\hat{k})-E(\hat{k}))}{\hat{k}}.$
(205)

###### Acknowledgements.

The authors thank FAPES/FAPEMIG/CNPq/CAPES for financial support. The authors
thank various anonymous referees for valuable comments that improved the
quality of the paper.

## Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated
or analysed during the current study.

## References

  * Gralla and Lupsasca [2020a] S. E. Gralla and A. Lupsasca, Lensing by kerr black holes, Physical Review D 101, 044031 (2020a).
  * Gralla  _et al._ [2020] S. E. Gralla, A. Lupsasca, and D. P. Marrone, The shape of the black hole photon ring: A precise test of strong-field general relativity, Physical Review D 102, 124004 (2020).
  * Gralla  _et al._ [2019] S. E. Gralla, D. E. Holz, and R. M. Wald, Black hole shadows, photon rings, and lensing rings, Physical Review D 100, 024018 (2019).
  * Collaboration  _et al._ [2019] E. H. T. Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. AZULY, _et al._ , First m87 event horizon telescope results. i. the shadow of the supermassive black hole, Astrophys. J. Lett 875, L1 (2019).
  * Akiyama  _et al._ [2022] K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, U. Bach, A.-K. Baczko, D. Ball, _et al._ , First sagittarius a* event horizon telescope results. i. the shadow of the supermassive black hole in the center of the milky way, The Astrophysical Journal Letters 930, L12 (2022).
  * Novikov and Thorne [1973] I. Novikov and K. Thorne, Les houches summer school of theoretical physics: Black holes (1973).
  * Ovalle  _et al._ [2021] J. Ovalle, E. Contreras, and Z. Stuchlik, Kerr–de sitter black hole revisited, Physical Review D 103, 084016 (2021).
  * Contreras  _et al._ [2021] E. Contreras, J. Ovalle, and R. Casadio, Gravitational decoupling for axially symmetric systems and rotating black holes, Physical Review D 103, 044020 (2021).
  * Ovalle [2017] J. Ovalle, Decoupling gravitational sources in general relativity: from perfect to anisotropic fluids, Physical Review D 95, 104019 (2017).
  * Ovalle [2019] J. Ovalle, Decoupling gravitational sources in general relativity: The extended case, Physics Letters B 788, 213 (2019).
  * Ovalle [2022] J. Ovalle, Warped vacuum energy by black holes, The European Physical Journal C 82, 1 (2022).
  * Hackmann  _et al._ [2010] E. Hackmann, C. Lämmerzahl, V. Kagramanova, and J. Kunz, Analytical solution of the geodesic equation in kerr-(anti-) de sitter space-times, Physical Review D 81, 044020 (2010).
  * Gralla and Lupsasca [2020b] S. E. Gralla and A. Lupsasca, Null geodesics of the kerr exterior, Physical Review D 101, 044032 (2020b).
  * Carter [1968] B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174, 1559 (1968).
  * Charbulák and Stuchlík [2017] D. Charbulák and Z. Stuchlík, Photon motion in kerr–de sitter spacetimes, The European Physical Journal C 77, 897 (2017).
  * Mino [2003] Y. Mino, Perturbative approach to an orbital evolution around a supermassive black hole, Physical Review D 67, 084027 (2003).
  * Teo [2003] E. Teo, Spherical photon orbits around a kerr black hole, General Relativity and Gravitation 35, 1909 (2003).
  * Omwoyo  _et al._ [2022] E. Omwoyo, H. Belich, J. C. Fabris, and H. Velten, Remarks on the black hole shadows in kerr-de sitter space times, The European Physical Journal C 82, 1 (2022).
  * Li  _et al._ [2020] P.-C. Li, M. Guo, and B. Chen, Shadow of a Spinning Black Hole in an Expanding Universe, Phys. Rev. D 101, 084041 (2020), arXiv:2001.04231 [gr-qc] .
  * Cunha and Herdeiro [2018] P. V. Cunha and C. A. Herdeiro, Shadows and strong gravitational lensing: a brief review, General Relativity and Gravitation 50, 1 (2018).
  * Bardeen [1973] J. M. Bardeen, Timelike and null geodesics in the Kerr metric, Proceedings, Ecole d’Eté de Physique Théorique: Les Astres Occlus : Les Houches, France, August, 1972, 215-240 , 215 (1973).
  * Cárdenas-Avendaño  _et al._ [2023] A. Cárdenas-Avendaño, A. Lupsasca, and H. Zhu, Adaptive analytical ray tracing of black hole photon rings, Phys. Rev. D 107, 043030 (2023), arXiv:2211.07469 [gr-qc] .
  * Ma  _et al._ [2022] S.-J. Ma, T.-C. Ma, J.-B. Deng, and X.-R. Hu, Black hole shadow, photon ring and lensing ring in the cdm halo, arXiv preprint arXiv:2206.12820 (2022).
  * Hou  _et al._ [2022] Y. Hou, P. Liu, M. Guo, H. Yan, and B. Chen, Multi-level images around Kerr–Newman black holes, Class. Quant. Grav. 39, 194001 (2022), arXiv:2203.02755 [gr-qc] .
  * Akiyama  _et al._ [2019] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, D. Bintley, _et al._ , First m87 event horizon telescope results. ii. array and instrumentation, The Astrophysical Journal Letters 875, L2 (2019).
  * Lima  _et al._ [2010] W. C. C. Lima, G. E. A. Matsas, and D. A. T. Vanzella, Awaking the vacuum in relativistic stars, Phys. Rev. Lett. 105, 151102 (2010), arXiv:1009.1771 [gr-qc] .
  * Tignol [2015] J.-P. Tignol,  _Galois’ theory of algebraic equations_ (World Scientific Publishing Company, 2015).
  * Fukushima [2012] T. Fukushima, Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations, Journal of Computational and Applied Mathematics 236, 1961 (2012).
  * Johnson  _et al._ [2020] M. D. Johnson, A. Lupsasca, A. Strominger, G. N. Wong, S. Hadar, D. Kapec, R. Narayan, A. Chael, C. F. Gammie, P. Galison, _et al._ , Universal interferometric signatures of a black hole’s photon ring, Science advances 6, eaaz1310 (2020).
  * Byrd and Friedman [1954] P. Byrd and M. Friedman, Handbook of elliptic integrals for engineers and physicists (ref. u. wegner), Mitteilungen der Astronomischen Gesellschaft Hamburg 5 (1954).
  * Kapec and Lupsasca [2019] D. Kapec and A. Lupsasca, Particle motion near high-spin black holes, Classical and Quantum Gravity 37, 015006 (2019).