For the rest of this section, we fix $(\mathcal{X},\mathfrak{S})$ to be an HHS
with the bounded domain dichotomy and unbounded products endowed with its ABD
structure, for which $S\in\mathfrak{S}$ is the $\sqsubset$-maximal domain. The
goal is to prove the reverse bound of Lemma 17.6 for some variation of our
cubical model construction, whose definition involves an idea from the world
of surfaces.

###  17.5. Motivation: Filling curve pairs

Let $\Sigma$ be a finite type surface and $\alpha,\beta$ simple closed curves.
Then $\alpha,\beta$ are _filling_ if, up to isotopy, the complement
$\Sigma-(\alpha\cup\beta)$ is a union of disks and once-punctured disks. Let
$\mathcal{C}(\Sigma)$ denote the curve graph of $\Sigma$, whose vertices are
isotopy classes of simple closed curves on $\Sigma$, with two classes
connected by an edge if and only if they have disjoint representatives. Then
$\alpha,\beta$ are filling if and only if
$d_{\mathcal{C}(\Sigma)}(\alpha,\beta)\geq 3$, because if $\gamma$ is a simple
closed curve disjoint (up to isotopy) from $\alpha\cup\beta$, then
$d_{\mathcal{C}(\Sigma)}(\alpha,\beta)\leq 2$ by the triangle inequality.

Suppose now that $\mu,\mu^{\prime}$ are a pair of (complete) markings on
$\Sigma$ (in the sense of [MM00]) which have sufficiently large projections to
the curve graphs associated to a family of subsurfaces $\mathcal{V}$ of
$\Sigma$. If $\alpha$ is any geodesic between the projections of
$\mu,\mu^{\prime}$ to $\mathcal{C}(\Sigma)$, then the Bounded Geodesic Image
Theorem [MM00, Theorem 3.1] says that $d_{\mathcal{C}{\Sigma}}(\alpha,\partial
V)\leq 1$ for any $V\in\mathcal{V}$. It is not hard to show that if in
addition the subsurfaces in $\mathcal{V}$ have pairwise filling boundaries,
then $\mathrm{diam}_{\mathcal{C}(\Sigma)}(\bigcup_{V\in\mathcal{V}}\partial
V)\asymp\\#\mathcal{V}$, since each has to be pairwise at least distance $3$
away, while also being distance $1$ from a geodesic.

In the rest of this section, we will first generalize this filling philosophy
to the setting of HHSes with unbounded products, and then tweak our cubical
models to encode this filling property into the structure of their
hyperplanes.

###  17.6. Filling pairs of domains

We begin with a definition that is motivated by the notion of a pair of
filling simple closed curves on a finite-type surface.

######  Definition 17.7 (Filling domains).

We say a pair of domains $U,V\in\mathfrak{S}$ is _filling_ if there does not
exist a domain $W\in\mathfrak{S}$ with $\mathrm{diam}(\mathcal{C}(W))=\infty$
so that $W\perp U$ and $W\perp V$.

Observe that if $U,V$ are not filling, then
$d_{S}(\rho^{U}_{S},\rho^{V}_{S})\leq 2E$. On the other hand:

######  Lemma 17.8.

Suppose $U,V\in\mathfrak{S}$ are filling and at least one of
$\mathrm{diam}(\mathcal{C}(U))=\infty$ or
$\mathrm{diam}(\mathcal{C}(V))=\infty$. Then if $W\in\mathfrak{S}$ with
$U\sqsubset W$ or $V\sqsubset W$, we have $W=S$.

###### Proof.

Suppose instead that $W\sqsubsetneq S$. Since at least one of
$\mathrm{diam}(\mathcal{C}(U))=\infty$ or
$\mathrm{diam}(\mathcal{C}(V))=\infty$, the Distance Formula 2.9 provides that
$\mathrm{diam}_{\mathcal{X}}(\mathbb{F}_{W})=\infty$, and hence
$\mathrm{diam}_{\mathcal{X}}(\mathbb{E}_{W})=\infty$ by our unbounded products
assumption. But then Lemma 17.3 implies that some $Z\perp W$ has
$\mathrm{diam}(\mathcal{C}(Z))=\infty$, which contradicts the assumption that
$U,V$ fill, giving us a contradiction. ∎

######  Lemma 17.9.

For any $K_{0}>\max\\{B,E\\}$, if $x,y\in\mathcal{X}$ and
$\mathcal{V}\subset\mathrm{Rel}_{K_{0}}(x,y)$ is a pairwise filling collection
of domains, then
$\\#\mathcal{V}\prec\mathrm{diam}_{S}(\bigcup_{V\in\mathcal{V}}\rho^{V}_{S})$.

###### Proof.

Note that by choosing $K_{0}>B$ (the bounded domain dichotomy constant,
Definition 17.1), we have that $\mathrm{diam}(\mathcal{C}(V))=\infty$ for all
$V\in\mathcal{V}$. Then the Large Links Axiom 6 provides domains
$T_{1},\dots,T_{N}\in\mathfrak{S}$ with $N\asymp d_{S}(x,y)$ so that each of
the domains in $\mathcal{V}$ nest into one of the $T_{i}$. On the other hand,
Lemma 17.8 says the only domain in which any pair of the domains in
$\mathcal{V}$ nests is $S$. Hence each $V\in\mathcal{V}$ nests into at most
one $T_{i}$, and thus $\\#\mathcal{V}\leq N\asymp d_{S}(x,y)$, as required. ∎

###  17.7. Well-separated cubical models

In this subsection, we explain how to modify the standard cubical model for
the hull of a pair of points $x,y\in\mathcal{X}$, so that any collection of
$0$-separated hyperplanes is labeled by a pairwise filling collection of
domains. The construction uses the setting of hierarchical family of trees as
in Definition 12.1, not in the HHS itself, as well as Tree Trimming 10.2.

Fix points $x,y\in\mathcal{X}$, their hull
$H=\mathrm{hull}_{\mathcal{X}}(x,y)$, and set of relevant domains
$\mathcal{U}=\mathrm{Rel}_{K}(x,y)$, where $K$ is the largeness constant as in
Section 5, nd also larger than the bounded domain dichotomy constant $B$ so
taht we can eventually apply Lemma 17.8. Let
$(\widehat{T}_{U})_{U\in\mathcal{U}}$ be the associated collection of
simplicial trees with their relative projections, as produced by Tree Trimming
10.2 as in Corollary 10.9.

We begin with the definition of extra domains which act as a firewall against
fake $0$-separation in the cubical model for $x,y$:

######  Definition 17.10 (Firewall domains).

For each $U\in\mathcal{U}$ with $U\neq S$, the $\sqsubset$-maximal domain, the
_firewall_ for $U$ is a domain label $V_{U}$ and a copy of the unit interval
$\widehat{T}_{V_{U}}=[0,1]$, along with the following relation and (collapsed)
relative projection data:

  * •

$V_{U}\sqsubset S$ and $\hat{\delta}^{S}_{V_{U}}\equiv 0$ is a constant map.

  * •

For each $W\in\mathcal{U}$ with $W,U$ not filling, we set $V_{U}\perp W$.

  * •

If $W\in\mathcal{U}-\\{S\\}$ forms a filling pair with $U$, then we set
$W\pitchfork V_{U}$, $\hat{\delta}^{V_{U}}_{W}=\hat{\delta}^{U}_{W}$ and
$\hat{\delta}^{W}_{U_{V}}=0$.

  * •

We set $\hat{x}_{V_{U}}=\hat{y}_{V_{U}}=0$.

######  Lemma 17.11.

The set of domain labels
$\mathcal{U}^{\prime}=\mathcal{U}\cup\\{V_{U}|U\in\mathcal{U}-\\{S\\}\\}$
along with its associated set of trees, projections, and relative projections,
is a hierarchical family of trees. Hence we can define its $0$-consistent set
$\mathcal{Q}^{\prime}$.

###### Proof.

We discuss the non-immediate items of Definition 12.1. Let
$U\in\mathcal{U}-\\{S\\}$. The BGI property follows from defining
$\hat{\delta}^{S}_{V_{U}}\equiv 0$. Finally, the tuples
$(\hat{x}_{U}),(\hat{y}_{U})$ are in $\mathcal{Q}^{\prime}$ because in each of
the addition $\widehat{T}_{V_{U}}$ domains, the coordinate coincides with all
possible relative projections (which all come from transverse relations). This
completes the proof.

∎

As proven in Theorem 13.6, the $0$-consistent $\mathcal{Q}^{\prime}$ is
isometric to a cube complex, and we refer to $\mathcal{Q}^{\prime}$ as the
_firewall cubical model_ for $\mathrm{hull}_{\mathcal{X}}(x,y)$. The following
explains the relationship between $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$:

######  Lemma 17.12.

The map $\Delta:\mathcal{Q}^{\prime}\to\mathcal{Q}$ which collapses each of
the new unit intervals $\widehat{T}_{V_{U}}$ for $U\in\mathcal{U}-\\{S\\}$ is
a uniform quasi-isometry, with constants depending only on $\mathcal{X}$.
Moreover, the map
$\Xi:\prod_{U\in\mathcal{U}}\widehat{T}_{U}\to\prod_{U\in\mathcal{U}^{\prime}}\widehat{T}_{U}$
which fixes coordinates for $U\in\mathcal{U}$ and assigns $0$ for the domains
in $\mathcal{U}^{\prime}-\mathcal{U}$ is an isometric embedding.

###### Proof.

The first statement is an immediate consequence of Tree Trimming 10.2. The
second statement is an immediate consequence of the observation that the only
new hyperplanes in $\mathcal{Q}^{\prime}$ that are not in $\mathcal{Q}$ are
labeled by domains in $\mathcal{U}^{\prime}-\mathcal{U}$, and these new
hyperplanes cannot separate vertices whose new coordinates are all the same by
Lemma 13.2. ∎

With this in hand, we can prove the following, which is the main point of the
construction:

######  Proposition 17.13.

A pair of $\mathcal{Q}^{\prime}$-hyperplanes $\mathbf{h},\mathbf{h}^{\prime}$
are $0$-separated if and only if their domain labels are filling.

###### Proof.

We first observe that

  1. (1)

Any hyperplane in $\mathcal{Q}^{\prime}$ labeled by $S$ crosses no other
hyperplane, and

  2. (2)

For every $U\in\mathcal{U}-\\{S\\}$, there is a hyperplane
$\mathbf{h}_{V_{U}}$ in $\mathcal{Q}^{\prime}$ labeled by $V_{U}$ which
crosses every hyperplane $\mathbf{h}$ which is labeled by a domain
$W\in\mathcal{U}^{\prime}$ for which $U,W$ is not a filling pair.

(1) is an immediate consequence of Proposition 12.18. (2) is an immediate
consequence of the construction, plus Lemma 12.13, which says that the
midpoint $\frac{1}{2}=h_{V_{U}}\in\widehat{T}_{V_{U}}=[0,1]$ determines a
nonempty hyperplane $\mathbf{h}_{V_{U}}$, and Proposition 12.18, which says
that $\mathbf{h}_{V_{U}}$ crosses any hyperplane labeled by a domain
orthogonal to $V_{U}$, which is, by definition, any domain not making a
filling pair with $U$.

With these observations in hand, first suppose that
$\mathbf{h},\mathbf{h}^{\prime}$ are $0$-separated, with domain labels
$U,W\in\mathcal{U}^{\prime}$ respectively. Then (1) and (2) above says that
$U,W$ are a filling a pair (with possibly one or both being $S$). On the other
hand, if $U,W$ are a filling pair, then we are done by Lemma 17.8 plus our
assumption that $(\mathcal{X},\mathfrak{S})$ satisfies the bounded domain
dichotomy. In particular, by choosing the largeness constant $K$ to be larger
than the constant $B$ in Definition 17.1, we guarantee that the set of domains
in $\mathcal{U}$ orthogonal to both $U,W$ is empty. Hence no hyperplanes can
cross both $\mathbf{h},\mathbf{h}^{\prime}$ by Lemma 12.10. This completes the
proof. ∎

We also need the following observation which is true of any cubical model in
any HHS $\mathcal{X}$:

######  Lemma 17.14.

Let $x,y\in\mathcal{X}$ be two points in any HHS and $\mathcal{Q}$ any cubical
model for the hull of $x,y$. For any $U\in\mathcal{U}$, if $m$ is the number
of $\mathcal{Q}$-hyperplanes labeled by $U$ separating $x$ from $y$, then
$m\prec d_{U}(x,y)$, where the constant of $\prec$ depends only on
$\mathcal{X}$.

###### Proof.

By item (4) of Lemma 7.14, we have $m\leq
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})\leq d_{T_{U}}(x_{U},y_{U})\prec
d_{U}(x,y)$, so we are done. ∎

We are now ready to prove the upper bound:

######  Proposition 17.15.

For any $x,y\in\mathcal{X}$ and the firewall cubical model
$\mathcal{Q}^{\prime}$ for $\mathrm{hull}_{\mathcal{X}}(x,y)$ satisfies

$d_{\mathcal{C}(S)}(x,y)\succ d^{\mathcal{Q}^{\prime}}_{0}(x,y),$

with constants depending only on $\mathcal{X}$.

###### Proof.

Let $\mathbf{h}_{1},\dots,\mathbf{h}_{n}$ be a $0$-separated chain separating
$\hat{x},\hat{y}\in\mathcal{Q}^{\prime}$. By Proposition 17.13 and Lemma 17.9,
the number $m$ of domains $U\sqsubsetneq S$ labeling hyperplanes in the chain
satisfies

$m\prec\mathrm{diam}_{S}(\bigcup_{V\in\mathcal{V}}\rho^{V}_{S})\prec
d_{S}(x,y),$

with the latter coarse inequality provided by the BGIA 7. On the other hand,
the number of domains in the chain labeled by $S$ is $n-m$. Hence we are done
by Lemma 17.9 and Lemma 17.14. ∎

## References

  * [ABD21] Carolyn Abbott, Jason Behrstock, and Matthew Durham.  Largest acylindrical actions and stability in hierarchically hyperbolsic groups.  Transactions of the American Mathematical Society, Series B, 8(3):66–104, 2021.
  * [ABR22] Carolyn Abbott, Jason Behrstock, and Jacob Russell.  Structure invariant properties of the hierarchically hyperbolic boundary.  arXiv preprint arXiv:2208.07930, 2022.
  * [Aou13] Tarik Aougab.  Uniform hyperbolicity of the graphs of curves.  Geometry & Topology, 17(5):2855–2875, 2013.
  * [BBF21] Mladen Bestvina, Ken Bromberg, and Koji Fujiwara.  Proper actions on finite products of quasi-trees.  Annales Henri Lebesgue, 4:685–709, 2021.
  * [BC12] Jason Behrstock and Ruth Charney.  Divergence and quasimorphisms of right-angled artin groups.  Mathematische Annalen, 352(2):339–356, 2012.
  * [BCM12] Jeffrey F Brock, Richard D Canary, and Yair N Minsky.  The classification of kleinian surface groups, ii: The ending lamination conjecture.  Annals of Mathematics, pages 1–149, 2012.
  * [Beh06a] Jason A Behrstock.  Asymptotic geometry of the mapping class group and teichmüller space.  Geometry & Topology, 10(3):1523–1578, 2006.
  * [Beh06b] Jason A Behrstock.  Asymptotic geometry of the mapping class group and teichmüller space.  Geometry & Topology, 10(3):1523–1578, 2006.
  * [Ber21] Daniel James Solomon Berlyne.  Hierarchical hyperbolicity of graph products and graph braid groups.  PhD thesis, City University of New York, 2021.
  * [BF21] Jonas Beyrer and Elia Fioravanti.  Cross-ratios on cat (0) cube complexes and marked length-spectrum rigidity.  Journal of the London Mathematical Society, 104(5):1973–2015, 2021\.
  * [BHMS20] Jason Behrstock, Mark Hagen, Alexandre Martin, and Alessandro Sisto.  A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups.  arXiv preprint arXiv:2005.00567, 2020.
  * [BHS17a] Jason Behrstock, Mark Hagen, and Alessandro Sisto.  Hierarchically hyperbolic spaces, i: Curve complexes for cubical groups.  Geometry & Topology, 21(3):1731–1804, 2017.
  * [BHS17b] Jason Behrstock, Mark F Hagen, and Alessandro Sisto.  Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups.  Proceedings of the London Mathematical Society, 114(5):890–926, 2017.
  * [BHS19] Jason Behrstock, Mark Hagen, and Alessandro Sisto.  Hierarchically hyperbolic spaces II: Combination theorems and the distance formula.  Pacific Journal of Mathematics, 299(2):257–338, 2019.
  * [BHS21] Jason Behrstock, Mark F Hagen, and Alessandro Sisto.  Quasiflats in hierarchically hyperbolic spaces.  Duke Mathematical Journal, 1(1):1–88, 2021.
  * [BKMM12] Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher.  Geometry and rigidity of mapping class groups.  Geometry & Topology, 16(2):781–888, 2012.
  * [BLMR20] Jeffrey Brock, Christopher Leininger, Babak Modami, and Kasra Rafi.  Limit sets of teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, ii.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2020(758):1–66, 2020.
  * [BM11] Jason A Behrstock and Yair N Minsky.  Centroids and the rapid decay property in mapping class groups.  Journal of the London Mathematical Society, 84(3):765–784, 2011\.
  * [Bow13] Brian Bowditch.  Coarse median spaces and groups.  Pacific Journal of Mathematics, 261, 02 2013.
  * [Bow16] Brian H Bowditch.  Some properties of median metric spaces.  Groups, Geometry, and Dynamics, 10(1):279–317, 2016.
  * [Bow17] Brian Bowditch.  Large-scale rigidity properties of the mapping class groups.  Pacific Journal of Mathematics, 293(1):1–73, 2017.
  * [Bow19] Brian H. Bowditch.  Convex hulls in coarse median spaces.  preprint, 2019.
  * [Bow22] Brian H Bowditch.  Median algebras (preliminary draft).  preprint, 2022.
  * [BR20] Federico Berlai and Bruno Robbio.  A refined combination theorem for hierarchically hyperbolic groups.  Groups, Geometry, and Dynamics, 14(4):1127–1203, 2020.
  * [Bri10] Martin R Bridson.  Semisimple actions of mapping class groups on cat (0) spaces.  Geometry of Riemann surfaces, 368:1–14, 2010.
  * [Bro03] Jeffrey Brock.  The weil-petersson metric and volumes of 3-dimensional hyperbolic convex cores.  Journal of the American Mathematical Society, 16(3):495–535, 2003\.
  * [CDH10] Indira Chatterji, Cornelia Druţu, and Frédéric Haglund.  Kazhdan and haagerup properties from the median viewpoint.  Advances in Mathematics, 225(2):882–921, 2010.
  * [Che00] Victor Chepoi.  Graphs of some cat (0) complexes.  Advances in Applied Mathematics, 24(2):125–179, 2000.
  * [Che22] Marissa Chesser.  Stable subgroups of the genus 2 handlebody group.  Algebraic & Geometric Topology, 22(2):919–971, 2022.
  * [CMW19] Jon Chaika, Howard Masur, and Michael Wolf.  Limits in pmf of teichmüller geodesics.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2019(747):1–44, 2019.
  * [CRHK21] Montserrat Casals-Ruiz, Mark Hagen, and Ilya Kazachkov.  Real cubings and asymptotic cones of hierarchically hyperbolic groups.  preprint, 2021.
  * [DDLS21a] Spencer Dowdall, Matthew G Durham, Christopher J Leininger, and Alessandro Sisto.  Extensions of veech groups ii: Hierarchical hyperbolicity and quasi-isometric rigidity.  To appear in _Commentarii Mathematici Helvetici_ ; arXiv:2111.00685, 2021.
  * [DDLS21b] Spencer Dowdall, Matthew G Durham, Christopher J Leininger, and Allesandro Sisto.  Extensions of veech groups i: A hyperbolic action.  To appear in the _Journal of Topology_ ; arXiv:2006.16425, 2021\.
  * [DHS17] Matthew Durham, Mark Hagen, and Alessandro Sisto.  Boundaries and automorphisms of hierarchically hyperbolic spaces.  Geometry & Topology, 21(6):3659–3758, 2017.
  * [DHS20a] Matthew Durham, Mark Hagen, and Alessandro Sisto.  Correction to the article boundaries and automorphisms of hierarchically hyperbolic spaces.  Geometry & Topology, 24(2):1051–1073, 2020.
  * [DHS20b] Matthew Durham, Mark Hagen, and Alessandro Sisto.  Correction to the article boundaries and automorphisms of hierarchically hyperbolic spaces.  Geometry & Topology, 24(2):1051–1073, 2020.
  * [DMS20] Matthew G Durham, Yair N Minsky, and Alessandro Sisto.  Stable cubulations, bicombings, and barycenters.  To appear in Geometry & Topology (arXiv:2009.13647), 2020.
  * [Dur16] Matthew Gentry Durham.  The augmented marking complex of a surface.  Journal of the London Mathematical Society, 94(3):933–969, 2016\.
  * [Dur23] Matthew Gentry Durham.  Semi-hyperbolicity of mapping class groups via stable cumulations.  In preparation, 2023.
  * [DZ22] Matthew Gentry Durham and Abdul Zalloum.  The geometry of genericity in mapping class groups and teichm$\backslash$” uller spaces via cat (0) cube complexes.  arXiv preprint arXiv:2207.06516, 2022.
  * [EMR17] Alex Eskin, Howard Masur, and Kasra Rafi.  Large-scale rank of teichmüller space.  Duke Mathematical Journal, 166(8):1517–1572, 2017.
  * [FFH23] Talia Fernós, David Futer, and Mark Hagen.  Homotopy equivalent boundaries of cube complexes.  arXiv preprint arXiv:2303.06932, 2023.
  * [Gen20] Anthony Genevois.  Hyperbolicities in CAT(0) cube complexes.  L’Enseignement Mathématique, 65(1):33–100, January 2020.
  * [Gro87] M. Gromov.  Hyperbolic groups.  In Essays in Group Theory, pages 75–263. Springer New York, 1987\.
  * [Hag13a] Mark F Hagen.  The simplicial boundary of a cat (0) cube complex.  Algebraic & Geometric Topology, 13(3):1299–1367, 2013.
  * [Hag13b] Mark F. Hagen.  Weak hyperbolicity of cube complexes and quasi-arboreal groups.  Journal of Topology, 7(2):385–418, 08 2013.
  * [Hag18] Mark F Hagen.  Corrigendum to the article the simplicial boundary of a cat (0) cube complex.  Algebr. Geom. Topol, 18(2):1251–1256, 2018.
  * [Ham09] Ursula Hamenstädt.  Geometry of the mapping class group ii: A biautomatic structure.  arXiv preprint arXiv:0912.0137, 2009.
  * [Ham21] Ursula Hamenstädt.  Geometry of the mapping class groups iia: Cat (0) cube complexes and boundaries.  Preprint available at www. math. uni-bonn. de/people/ursula/noncube. pdf, page 34, 2021.
  * [HHP20] Thomas Haettel, Nima Hoda, and Harry Petyt.  Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity.  To appear in _Geometry & Topology_; arXiv preprint arXiv:2009.14053, 2020.
  * [HMS21] Mark Hagen, Alexandre Martin, and Alessandro Sisto.  Extra-large type artin groups are hierarchically hyperbolic.  arXiv preprint arXiv:2109.04387, 2021.
  * [HP98] Frédéric Haglund and Frédéric Paulin.  Simplicité de groupes d’automorphismes d’espaces à courbure négative.  Université de Paris-Sud. Département de Mathématique, 1998\.
  * [HP23a] Mark F Hagen and Harry Petyt.  Personal communication.  2023\.
  * [HP23b] Mark F Hagen and Harry Petyt.  Projection complexes and quasimedian maps.  Algebraic & Geometric Topology, 22(7):3277–3304, 2023.
  * [HRSS22] Mark Hagen, Jacob Russell, Alessandro Sisto, and Davide Spriano.  Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms.  arXiv preprint arXiv:2206.12244, 2022.
  * [HS20] Mark F Hagen and Tim Susse.  On hierarchical hyperbolicity of cubical groups.  Israel Journal of Mathematics, 236(1):45–89, 2020.
  * [HW14] G Christopher Hruska and Daniel T Wise.  Finiteness properties of cubulated groups.  Compositio Mathematica, 150(3):453–506, 2014.
  * [IMZ21] Merlin Incerti-Medici and Abdul Zalloum.  Sublinearly morse boundaries from the viewpoint of combinatorics.  arXiv preprint arXiv:2101.01037, 2021.
  * [KK14] Sang-Hyun Kim and Thomas Koberda.  The geometry of the curve graph of a right-angled artin group.  International Journal of Algebra and Computation, 24(02):121–169, 2014.
  * [LLR18] Christopher Leininger, Anna Lenzhen, and Kasra Rafi.  Limit sets of teichmüller geodesics with minimal non-uniquely ergodic vertical foliation.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2018(737):1–32, 2018.
  * [Min96] Yair N. Minsky.  Extremal length estimates and product regions in Teichmüller space.  Duke Math. J., 83(2):249–286, 1996.
  * [Min10] Yair Minsky.  The classification of kleinian surface groups, i: Models and bounds.  Annals of Mathematics, pages 1–107, 2010.
  * [MM99] Howard A Masur and Yair N Minsky.  Geometry of the complex of curves I: Hyperbolicity.  Inventiones mathematicae, 138(1):103–149, 1999.
  * [MM00] Howard A Masur and Yair N Minsky.  Geometry of the complex of curves II: Hierarchical structure.  Geometric and Functional Analysis, 10(4):902–974, 2000.
  * [Mou19] Sarah C Mousley.  Exotic limit sets of teichmüller geodesics in the hhs boundary.  Groups, Geometry, and Dynamics, 13(3):1057–1081, 2019.
  * [NR98] Graham A Niblo and Lawrence D Reeves.  The geometry of cube complexes and the complexity of their fundamental groups.  Topology, 37(3):621–633, 1998.
  * [Pet21] Harry Petyt.  Mapping class groups are quasicubical.  arXiv preprint arXiv:2112.10681, 2021.
  * [PS20] Harry Petyt and Davide Spriano.  Unbounded domains in hierarchically hyperbolic groups.  To appear in _Groups, Geometry, and Dynamics_ ; arXiv:2007.12535, 2020.
  * [PSZ22] Harry Petyt, Davide Spriano, and Abdul Zalloum.  Hyperbolic models for cat (0) spaces.  arXiv preprint arXiv:2207.14127, 2022.
  * [PSZ23] Harry Petyt, Davide Spriano, and Abdul Zalloum.  Personal communication.  2023\.
  * [QRT20] Yulan Qing, Kasra Rafi, and Giulio Tiozzo.  Sublinearly morse boundary ii: Proper geodesic spaces, 2020.
  * [Raf07] Kasra Rafi.  A combinatorial model for the teichmüller metric.  GAFA Geometric And Functional Analysis, 17(3):936–959, 2007.
  * [Raf14] Kasra Rafi.  Hyperbolicity in teichmüller space.  Geometry & Topology, 18(5):3025–3053, 2014.
  * [Rol16] Martin Roller.  Poc sets, median algebras and group actions.  arXiv preprint arXiv:1607.07747, 2016.
  * [RS09] Kasra Rafi and Saul Schleimer.  Covers and the curve complex.  Geometry & Topology, 13(4):2141–2162, 2009.
  * [RST18] Jacob Russell, Davide Spriano, and Hung Cong Tran.  Convexity in hierarchically hyperbolic spaces.  To appear in Algebraic & Geometric Topology (arXiv:1809.09303), 2018.
  * [Rus21] Jacob Russell.  Extensions of multicurve stabilizers are hierarchically hyperbolic.  arXiv preprint arXiv:2107.14116, 2021.
  * [Sag95] Michah Sageev.  Ends of group pairs and non-positively curved cube complexes.  Proceedings of the London Mathematical Society, 3(3):585–617, 1995\.
  * [Sag14] Michah Sageev.  $\mathrm{CAT(0)}$ cube complexes and groups.  In Geometric group theory, volume 21 of IAS/Park City Math. Ser., pages 7–54. Amer. Math. Soc., Providence, RI, 2014.
  * [She22] Sam Shepherd.  A cubulation with no factor system.  arXiv preprint arXiv:2208.10421, 2022.
  * [Sis17] Alessandro Sisto.  What is a hierarchically hyperbolic space?  arXiv preprint arXiv:1707.00053, 2017.
  * [ST19] Alessandro Sisto and Samuel J Taylor.  Largest projections for random walks and shortest curves in random mapping tori.  Mathematical Research Letters, 26(1):293–321, 2019.
  * [Ver93] Eric Robert Verheul.  Multimedians in metric and normed spaces.  CWI Tracts, 1993.
  * [Zal23a] Abdul Zalloum.  Effective flipping, skewering and rank rigidity for cubulated groups with factor systems.  arXiv preprint arXiv:2305.16298, 2023.
  * [Zal23b] Abdul Zalloum.  Injectivity, cubical approximations and equivariant wall structures beyond cat (0) cube complexes.  arXiv preprint arXiv:2305.02951, 2023.
  * [Zal23c] Abdul Zalloum.  Injectivity, cubical approximations and equivariant wall structures beyond cat (0) cube complexes.  arXiv preprint arXiv:2305.02951, 2023.


# On Schur Rings over Infinite Groups III

Nicholas Bastian111Nicholas Bastian, Southern Utah University,
<EMAIL_ADDRESS>Andrew Misseldine222Corresponding Author: Andrew
Misseldine, Southern Utah University<EMAIL_ADDRESS>(telephone)
435-865-8228, (fax) 435-865-8666

###### Abstract

In the paper, we develop further the properties of Schur rings over infinite
groups, with particular emphasis on the virtually cyclic group
$\mathcal{Z}\times\mathcal{Z}_{p}$, where $p$ is a prime. We provide structure
theorems for primitive sets in these Schur rings. In the case of Fermat and
safe primes, a complete classification theorem is proven which states that all
Schur rings over $\mathcal{Z}\times\mathcal{Z}_{p}$ are traditional. We also
draw analogs between Schur rings over $\mathcal{Z}\times\mathcal{Z}_{p}$ and
partitions of difference sets over $\mathcal{Z}_{p}$.

Keywords: Schur ring, group ring, cyclic group, virtually cyclic group,
association scheme, difference set

MSC Classification: 20c07, 16s34, 20e22, 05e30, 05b10

##  1 Introduction

The paper continues developing the theory of Schur rings over infinite groups
begun in [1, 3]. Schur rings (or S-rings) were first studied by Wielandt [9]
but only in the case of finite groups. Schur rings are of great interest in
algebraic combinatorics because of their connections to association schemes,
Cayley graphs, difference sets, and related objects. Let $\mathcal{Z}=\langle
z\rangle$ be the infinite cyclic group, written multiplicatively, and, for any
positive integer $n$, let $\mathcal{Z}_{n}=\langle a\rangle$ be the cyclic
group of order $n$, also written multiplicatively. In [1], it was shown that
there are exactly two Schur rings over $\mathcal{Z}$ and countably many Schur
rings over $\mathcal{Z}\times\mathcal{Z}_{2}$ belonging to one of four types.
This paper will extend these results to $\mathcal{Z}\times\mathcal{Z}_{n}$.

We remind the reader of some important notation and terminology introduced in
[1]. Let $F[G]$ denote the group algebra over group $G$ and field of
coefficients $F$, which we assume has characteristic $0$. Suppose
$\alpha=\sum_{g\in G}\alpha_{g}g\in F[G]$. Then define
$\alpha^{\ast}=\sum_{g\in G}\alpha_{g}g^{-1}$, which gives an involution on
$F[G]$, and the _Hadamard product_ $\alpha\circ\beta=\sum_{g\in
G}\alpha_{g}\beta_{g}g$, which gives a commutative, associative, bilinear
multiplication on $F[G]$. For any finite subset $C\subseteq G$, define
$\overline{C}=\sum_{g\in C}g$, called a _simple quantity_. Similarly, let
$C^{*}$ denote the set of inverses of elements of $C$. Note that for subsets
$C,D\subseteq G$, we have $\overline{C}\circ\overline{D}=\overline{C\cap D}$
and, if $C$ and $D$ are disjoint, $\overline{C}+\overline{D}=\overline{C\cup
D}$.

Let $\mathcal{D}$ be a partition of the group $G$ of finite support, that is,
if $C\in\mathcal{D}$ then $|C|<\infty$. Let
$\mathfrak{S}=\text{Span}_{F}\\{\overline{C}\mid C\in\mathcal{D}\\}\subseteq
F[G]$. We say that $\mathfrak{S}$ is a _Schur ring_ if additionally

  1. (i)

$\\{1\\}\in\mathcal{D}$

  2. (ii)

if $C\in\mathcal{D}$, then $C^{*}\in\mathcal{D}$

  3. (iii)

for all $C,D\in\mathcal{D}$,
$\overline{C}\,\overline{D}=\sum_{E\in\mathcal{D}}\lambda_{CDE}\overline{E}$,
where all but finitely many $\lambda_{CDE}$ are equal to 0. (The coefficient
$\lambda_{CDE}$ is called the _multiplicity_ of $\overline{E}$ in the product
$\overline{C}\,\overline{D}$.)

Note that Schur rings are subrings of $F[G]$ closed under $\circ$ and $*$.

Leung and Man [4, 5] classified all Schur rings over cyclic groups of finite
order. Particularly, they showed that all Schur rings over finite cyclic
groups are either trivial, automorphic, direct products, or wedge products
(see Section 2 for these definitions). We call Schur rings _traditional_ if it
is one of these four types. Similarly, we say that a group $G$ is
_traditional_ if all of its Schur rings are traditional. Hence, Leung and Man
showed that finite cyclic groups are traditional. In [1], it was shown that
$\mathcal{Z}$ and $\mathcal{Z}\times\mathcal{Z}_{2}$ were traditional, among
other infinite groups. In the case of $\mathcal{Z}$, there are only two Schur
rings, both automorphic, namely the discrete and symmetric Schur rings
$F[\mathcal{Z}]$ and $F[\mathcal{Z}]^{\pm}$, respectively. For
$\mathcal{Z}\times\mathcal{Z}_{2}$, it was shown that all Schur rings are of
one of the following forms:

  1. (i)

$F[\mathcal{Z}_{2}]\wedge F[\mathcal{Z}]$ or $F[\mathcal{Z}_{2}]\wedge
F[\mathcal{Z}]^{\pm}$,

  2. (ii)

$F[H\times\mathcal{Z}_{2}]\wedge F[\mathcal{Z}]$ or
$F[H\times\mathcal{Z}_{2}]^{\pm}\wedge F[\mathcal{Z}]^{\pm}$,

  3. (iii)

$F[H\times\mathcal{Z}_{2}]^{\langle\psi\rangle}\wedge F[\mathcal{Z}]^{\pm}$ or
$F[\mathcal{Z}\times\mathcal{Z}_{2}]^{\langle\psi\rangle}$,

  4. (iv)

$F[\mathcal{Z}\times\mathcal{Z}_{2}]$ or
$F[\mathcal{Z}\times\mathcal{Z}_{2}]^{\pm}$,

where $1<H\leq\mathcal{Z}$ and
$\psi:\mathcal{Z}\times\mathcal{Z}_{2}\to\mathcal{Z}\times\mathcal{Z}_{2}$ is
the automorphism induced by the relation $\psi:z\mapsto az^{-1}$, $a\mapsto
a$.

In this paper, we prove an analogous result for the group
$\mathcal{Z}\times\mathcal{Z}_{p}$, where $p$ belongs to one of two families
of odd primes, namely the Fermat and safe primes. A prime $p$ is called a
_Fermat prime_ [7] if $p=2^{k}+1$. Necessarily, it must be that $k=2^{\ell}$,
that is, $p=2^{2^{\ell}}+1$. There are only five known Fermat primes, namely
$3$, $5$, $17$, $257$, and $65537$. It is conjectured that these are the only
Fermat primes. We say a prime $p$ is a _safe prime_ [8] if $p=2q+1$ where $q$
is itself a prime number.333The associated prime $q$ to the safe prime
$p=2q+1$ is called a _Sophie Germain prime_. Safe primes received their name
because of their historical usage in cryptography. The first few safe primes
are $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$. It is conjectured
that there are infinitely many safe primes.

We now present the main result.

######  Theorem 1.1.

Let $p$ be a Fermat or safe prime. Then all Schur rings over the virtually
cyclic group $\mathcal{Z}\times\mathcal{Z}_{p}$ are of one of the following
forms:

  1. (i)

$F[\mathcal{Z}_{p}]^{\mathcal{H}}\wedge F[\mathcal{Z}]$ or
$F[\mathcal{Z}_{p}]^{\mathcal{H}}\wedge F[\mathcal{Z}]^{\pm}$,

  2. (ii)

$F[H\times\mathcal{Z}_{p}]^{\mathcal{H}}\wedge F[\mathcal{Z}]$ or
$F[H\times\mathcal{Z}_{p}]^{\mathcal{H}}\wedge F[\mathcal{Z}]^{\pm}$,

  3. (iii)

$F[\mathcal{Z}\times\mathcal{Z}_{p}]^{\mathcal{H}}$,

where $H\leq\mathcal{Z}$ and
$\mathcal{H}\leq\operatorname*{Aut}(\mathcal{Z}\times\mathcal{Z}_{p})\cong
GA(1,p)\times\mathcal{Z}_{2}$, where $GA(1,p)$ denotes the general affine
group over a finite vector space of order $p$. Hence,
$\mathcal{Z}\times\mathcal{Z}_{p}$ is traditional and has countably many Schur
rings.

The proof of Theorem 1.1 will be found in Section 4. Section 2 contains
important properties of Schur rings over infinite groups relevant to this
proof. Section 3 considers general structure theorems about primitive sets of
Schur rings over $\mathcal{Z}\times\mathcal{Z}_{n}$ and
$\mathcal{Z}\times\mathcal{Z}_{p}$, where $n$ is any positive integer and $p$
is a prime. Section 5 concludes with remarks relating Schur rings over
$\mathcal{Z}\times\mathcal{Z}_{n}$ with partitions of difference sets.

Acknowledgements: All calculations made in preparation of this paper were made
using MAGMA [2]. The authors are grateful to Stephen Humphries for useful
conversations and to the anonymous referees for their most helpful comments.

##  2 Properties of Schur Rings

In this section, we gather important terminology and properties about Schur
rings in general that will be useful for forthcoming proofs. Many of these
were known to Wielandt [10] in his early work, although they have been
extended to the infinite case. Most of these properties will be mentioned
without proof.444These omitted proofs can be found in [1].

The partition associated to a Schur ring $\mathfrak{S}$ will be denoted
$\mathcal{D}(\mathfrak{S})$. The elements of $\mathcal{D}(\mathfrak{S})$ are
called the _$\mathfrak{S}$ -classes_ (or _primitive sets_ of $\mathfrak{S}$).
Essentially, an element of the group algebra belongs to a Schur ring
$\mathfrak{S}$ only if it has constant coefficients across each
$\mathfrak{S}$-class. We say that a subset $C\subset G$ is an _$\mathfrak{S}$
-subset_ if $C$ is a union of the $\mathfrak{S}$-classes. When $G$ is finite
this is equivalent to $\overline{C}\in\mathfrak{S}$. We say a subset
$H\subseteq G$ is an _$\mathfrak{S}$ -subgroup_ if $H$ is an
$\mathfrak{S}$-subset and a subgroup of $G$. For any $\mathfrak{S}$-subgroup
$H$, let $\mathfrak{S}_{H}:=\mathfrak{S}\cap F[H]$, called a _Schur subring_.
For any Schur ring, the set of $\mathfrak{S}$-subsets forms a lattice closed
under intersections and unions and the set of Schur subrings forms a lattice
closed under intersections and joins. The associated partitions to these Schur
subrings are the common coarsenings and the common refinements of
$\mathcal{D}(\mathfrak{S}_{H})$ and $\mathcal{D}(\mathfrak{S}_{K})$, denoted
$\mathfrak{S}_{H\cap K}$ and $\mathfrak{S}_{\langle H,K\rangle}$,
respectively.

For an element $\alpha\in F[G]$, let
$\operatorname{supp}(\alpha):=\\{g\mid\alpha_{g}\neq{0}\\}\subseteq G$, which
is called the _support_ of $\alpha$ and must necessarily be finite. If
$\mathfrak{S}$ is a Schur ring over group $G$, $\alpha\in\mathfrak{S}$, and
$G_{\alpha}=\\{g\in G\mid\alpha g=\alpha\\}$ is the stabilizer subgroup, then
$G_{\alpha}$ is an $\mathfrak{S}$-subgroup of $G$. When $\alpha=\overline{C}$
is a simple quantity, we will denote $G_{\alpha}$ as $G_{C}$. Also, if
$H=\langle\operatorname{supp}(\alpha)\rangle$, then $H$ is an
$\mathfrak{S}$-subgroup.

Define the _$n$ th Frobenius map_ for any integer $n$ by the rule
$\alpha^{(n)}:=\sum_{g\in{G}}\alpha_{g}g^{n}$ whenever $\alpha=\sum_{g\in
G}\alpha_{g}g$. The Frobenius map is a bilinear map $F[G]\to F[G]$ such that
$\alpha^{(mn)}=\left(a^{(m)}\right)^{(n)}$ for any integers $n$ and $m$ and
$\alpha^{(-1)}=\alpha^{*}$. We define the Frobenius map on subsets of $G$
analogously. For example, all the subgroups of $\mathcal{Z}$ can be written as
$\mathcal{Z}^{(n)}$ for some integer $n$. The Frobenius map is very useful in
determining the structure of primitive sets of Schur rings over abelian
groups. In fact, if $\mathfrak{S}$ is a Schur ring over an abelian group $G$
and $m$ is an integer coprime to the orders of all torsion elements of $G$,
then for all $\alpha\in\mathfrak{S}$ we have that
$\alpha^{(m)}\in\mathfrak{S}$. Using the Frobenius map, we see that for a
Schur ring $\mathfrak{S}$ over an abelian group $G$, if the torsion subgroup
$T(G)$ has finite exponent then $T(G)$ is an $\mathfrak{S}$-subgroup. Hence,
Schur rings over abelian groups have a torsion Schur subring, denoted
$T(\mathfrak{S})$.

Let $\varphi:G\to H$ be a group homomorphism. Then this map linearly lifts to
the group ring in a natural way and will be denoted by the same symbol
$\varphi:F[G]\to F[H]$. A ring homomorphism between group rings of this form
is called a _Cayley homomorphism_ (the Frobenius map is an example of such).
If $\mathfrak{S}$ is a Schur ring over $G$ and additional $\ker\varphi$ is an
$\mathfrak{S}$-subgroup, then $\varphi(\mathfrak{S})$ is a Schur ring over
$\varphi(G)$. In particular,
$\mathcal{D}(\varphi(\mathfrak{S}))=\\{\varphi(C)\mid
C\in\mathcal{D}(\mathfrak{S})\\}$. Additionally, if
$C\in\mathcal{D}(\mathfrak{S})$, then the non-zero intersection numbers with
$C$ and the cosets of $\ker\varphi$ are constant, that is, $|C\cap
g(\ker\varphi)|=|C\cap g^{\prime}(\ker\varphi)|$ whenever these intersections
are non-empty. Furthermore, if two $\mathfrak{S}$-classes both intersect some
coset of $K$, then they intersect all the same cosets of $K$. These facts
imply that $\mathfrak{S}/T(\mathfrak{S})$ is a Schur ring over $G/T(G)$ for
any abelian group $G$.

Frobenius maps, torsion subgroups, and Cayley projections will prove to be
helpful in determining the structure of the primitive sets of Schur rings over
$\mathcal{Z}\times\mathcal{Z}_{n}$. We introduce two other important counting
arguments that were not included in [1] but which will also prove useful here.

The first counting argument is actually two lemmas about the lengths of
primitive sets used by Scott [6] for which we extend to infinite groups.

######  Lemma 2.1 ([6] 13.8.2).

Let $C$, $D$, $E\in\mathcal{D}(\mathfrak{S})$ be primitive sets of a Schur
ring $\mathfrak{S}$. Suppose that $\overline{E}$ appears in the product
$\overline{C}\,\overline{D}$ with nonzero multiplicity $\lambda$. Likewise,
let $\mu$ and $\nu$ be the multiplicities of $\overline{C}^{*}$ and
$\overline{D}^{*}$ in the products $\overline{D}\,\overline{E}^{*}$ and
$\overline{E}^{*}\overline{C}$, respectively. Then $\lambda|E|=\mu|C|=\nu|D|$.

In other words, for any $\mathfrak{S}$-classes $C$, $D$, $E$, it holds that

$\lambda_{CDE}|E|=\lambda_{DE^{*}C^{*}}|C|=\lambda_{E^{*}CD^{*}}|D|.$

######  Lemma 2.2 ([6] 13.8.3).

Let $\mathfrak{S}$ be a Schur ring. Let $C,D\in\mathcal{D}(\mathfrak{S})$ be
primitive sets such that $\gcd(|C|,|D|)=1$. Then
$\overline{C}\,\overline{D}=\lambda\overline{E}$ for some primitive set
$E\in\mathcal{D}(\mathfrak{S})$.

As the proofs are the same as found in [6] without modification, they are
omitted. It should be noted that some assumptions made by Scott are omitted in
these lemmas, particularly the assumption that the Schur ring be _primitive_
,555We say that a Schur ring $\mathfrak{S}$ is _primitive_ if it has no
nontrivial, proper $\mathfrak{S}$-subgroups (this definition, while
equivalent, is not the definition originally used by Wielandt or Scott). The
primitive Schur rings have been studied extensively, particularly because of
their applications to permutation groups. In [6], Scott uses 13.8.2 and 13.8.3
to rule out the existence of primitive Schur rings over groups of small order
in a way analogous to how Sylow theorems are used to show the non-existence of
simple groups of small order. as they are not actually used in the proof of
13.8.2 and the portion of 13.8.3 which requires them is not included in Lemma
2.2 above.

For the second counting argument, let $A,B$ be finite subsets of a group $G$
such that $|A|=|B|$. We say that an element $x\in AB^{*}$ is a _tycoon_ 666A
_difference set_ $D$ is a subset of a group $G$ such that the multiplicity of
every non-identity element in $G$ appears in $DD^{*}$ with the same
multiplicity. When $G$ is finite, this is equivalent to
$\overline{D}\,\overline{D}^{*}=n+\lambda\overline{G}$. Thus, difference sets
guarantee the greatest equity among the multiplicities of group elements in
the product $DD^{*}$. As the multiplicity of any element in the product
$AB^{*}$ ranges between $0$ and $|A|$, a set containing a tycoon represents
the greatest possible inequity between multiplicities of group elements. If
multiplicities are replaced by wealth for the sake of analogy, the curious
name tycoon is then explained. in $(A,B)$ if the multiplicity of $x$ in
$AB^{*}$ is $|A|$. Note that if $x$ is a tycoon in $(A,B)$ then $x^{-1}$ is a
tycoon in $(B,A)$.

The simplest example of a tycoon is the group identity in the pair $(A,A)$. In
fact, every tycoon $x$ in $(A,B)$ is essentially just this example up to
translation. More specifically, $x$ is a tycoon in $(A,B)$ if and only if
$A=xB$ if and only if $AB^{*}=xBB^{*}=AA^{*}x$. This can be seen by counting
solutions to the equation $ab^{-1}=x$ for $a\in A$, $b\in B$ and a fixed
tycoon $x\in AB^{*}$.

While the existence of tycoons is fairly trivial, the existence of two tycoons
in an $\mathfrak{S}$-class will be useful. After all, if a pair $(A,B)$ has
two tycoons, say $x$, $y$, then $A=xB=yB$. Thus, $B$ is stabilized by
$x^{-1}y$, which implies that $G_{B}\neq 1$. When $A$ and $B$ are
$\mathfrak{S}$-subsets, multiple tycoons will provide nontrivial
$\mathfrak{S}$-subgroups. We summarize this in the following lemma.

######  Lemma 2.3.

Let $\mathfrak{S}$ be a Schur ring with finite $\mathfrak{S}$-subsets $A$, $B$
such that $|A|=|B|$. If the pair $(A,B)$ has at least two distinct tycoons,
then $A$ and $B$ are both unions of cosets of some (necessarily finite)
nontrivial $\mathfrak{S}$-subgroup.

We briefly remind the reader of the four types of traditional Schur rings. For
a finite group $G$, the partition
$\mathcal{D}(\mathfrak{S})=\\{1,G\smallsetminus 1\\}$ always affords a Schur
ring, called the _trivial Schur ring_. On the other extreme, for any group $G$
(finite or infinite) the partition of singletons affords a Schur ring, known
as the _discrete Schur ring_. This Schur ring coincides with the group algebra
itself.

If $\mathcal{H}\leq\operatorname*{Aut}(G)$ is a finite automorphism subgroup,
then the set of elements of $F[G]$ fixed by $\mathcal{H}$ is a Schur ring over
$G$, denoted $F[G]^{\mathcal{H}}$ and called the _automorphic Schur ring_ (or
_orbit Schur ring_) associated to $\mathcal{H}$. The discrete Schur ring is
automorphic where $\mathcal{H}=1$. When $G$ is abelian, the automorphic Schur
ring associated to $\mathcal{H}=\langle*\rangle$, consisting of inverse pairs,
is called the _symmetric Schur ring_ and is denoted $F[G]^{\pm}$.

Suppose $G=H\times K$. If $\mathfrak{S}$ and $\mathfrak{T}$ are Schur rings
over $H$ and $K$, respectively, then the ring
$\mathfrak{S}\otimes_{F}\mathfrak{T}$ has a Schur ring structure, called the
_direct product_ of $\mathfrak{S}$ and $\mathfrak{T}$ and is denoted
$\mathfrak{S}\times\mathfrak{T}$. The associated partition is
$\mathcal{D}(\mathfrak{S}\times\mathfrak{T})=\\{CD\mid
C\in\mathcal{D}(\mathfrak{S}),D\in\mathcal{D}(\mathfrak{T})\\}$. Also,
$F[H]^{\mathcal{H}}\times F[K]^{\mathcal{K}}=F[H\times
K]^{\mathcal{H}\times\mathcal{K}}$, that is, the direct product of automorphic
Schur rings is automorphic.

The final family of traditional Schur rings is due to Leung and Man [5]. Let
$H,K\leq G$ be two nontrivial, proper subgroups such that $K$ is finite,
$K\leq H$, and $K\trianglelefteq G$. Let $\mathfrak{S}$ be a Schur ring over
$H$ with $K$ as an $\mathfrak{S}$-subgroup. Suppose $\varphi:G\to G/K$ is the
natural quotient map. Then $\varphi(\mathfrak{S})$ is a Schur ring over $G/K$.
Let $\mathfrak{T}$ be a Schur ring over $G/K$ such that
$\mathfrak{T}_{H/K}=\varphi(\mathfrak{S})$. Then define the _wedge product_
$\mathfrak{S}\wedge\mathfrak{T}$ by the partition

$\mathcal{D}(\mathfrak{S}\wedge\mathfrak{T})=\mathcal{D}(\mathfrak{S})\cup\\{\varphi^{-1}(D)\mid
D\in\mathcal{D}(\mathfrak{T})\smallsetminus\mathcal{D}(\mathfrak{T}_{H/K})\\}.$

Under these conditions, $\mathfrak{S}\wedge\mathfrak{T}$ is a Schur ring over
$G$, called the _wedge product_ of $\mathfrak{S}$ and $\mathfrak{T}$.
Alternatively, a Schur ring $\mathfrak{S}$ over $G$ is a wedge product if
there exist nontrivial, proper $\mathfrak{S}$-subgroups $H,K\leq G$ such
$K\leq H$, $K\trianglelefteq G$, and every $\mathfrak{S}$-class outside of $H$
is a union of $K$-cosets (this necessarily implies that $K$ must be finite).
In this case, we say that $1<K\leq H<G$ is a _wedge-decomposition_ of
$\mathfrak{S}$.

##  3 The Structure of Primitive Sets of Schur Rings over
$\mathcal{Z}\times\mathcal{Z}_{n}$

We now consider the virtually cyclic group
$G=\mathcal{Z}\times\mathcal{Z}_{n}=\langle z\rangle\times\langle a\rangle$.
Let $\mathfrak{S}$ be a Schur ring over $G$. As mentioned above,
$T(G)=\mathcal{Z}_{n}$ is an $\mathfrak{S}$-subgroup. Hence, $T(\mathfrak{S})$
is some Schur subring over $\mathcal{Z}_{n}$, to which Leung’s and Man’s
classification theorem applies. Let $\varphi:G\to\mathcal{Z}$ be the natural
projection map. Again, $\ker\varphi=\mathcal{Z}_{n}$ is an
$\mathfrak{S}$-subgroup. Hence, $\varphi(\mathfrak{S})$ is a Schur ring over
$\mathcal{Z}$. As there are only two such Schur rings, the discrete and
symmetric rings, $\varphi(\mathfrak{S})$ is equal to one of these. Many of the
following proofs will be divided into one of two cases based upon the image
$\varphi(\mathfrak{S})$. For example, if $H\leq\mathcal{Z}$ is an
$\mathfrak{S}$-subgroup, then the Schur subring $\mathfrak{S}_{H}$ maps
isomorphically onto $\varphi(\mathfrak{S}_{H})\leq F[\mathcal{Z}]$. If $H\neq
1$, then $H\cong\mathcal{Z}$, and, hence, $\varphi(\mathfrak{S}_{H})$ itself
is either discrete or symmetric. The structure of $\varphi(\mathfrak{S}_{H})$
must agree with $\varphi(\mathfrak{S})$, that is, they are either both
discrete or both symmetric. As
$\mathfrak{S}_{H}\cong\varphi(\mathfrak{S}_{H})$, $\mathfrak{S}_{H}$ is
discrete when $\varphi(\mathfrak{S})$ is discrete and symmetric when
$\varphi(\mathfrak{S})$ is symmetric for all nontrivial
$\mathfrak{S}$-subgroups $H\leq\mathcal{Z}$.

For another example of these two cases, let $t$ be any integer. If
$\varphi(\mathfrak{S})$ is discrete then $\\{z^{t}\\}$ is an
$\varphi(\mathfrak{S})$-class. Then
$\varphi^{-1}(\\{z^{t}\\})=z^{t}\mathcal{Z}_{n}$ is an $\mathfrak{S}$-subset
(but not necessarily primitive). Likewise, if $\varphi(\mathfrak{S})$ is
symmetric then $\\{z^{t},z^{-t}\\}$ is a $\varphi(\mathfrak{S})$-class. Then
$\varphi^{-1}(\\{z^{t},z^{-t}\\})=z^{t}\mathcal{Z}_{n}\cup
z^{-t}\mathcal{Z}_{n}$ is an $\mathfrak{S}$-subset. In either case,
$\\{z^{t},z^{-t}\\}\mathcal{Z}_{n}$ is an $\mathfrak{S}$-subset for all
integers $t$, which generates its supporting $\mathfrak{S}$-subgroup $\langle
z^{t}\rangle\times\mathcal{Z}_{n}=\mathcal{Z}^{(t)}\times\mathcal{Z}_{n}\cong\mathcal{Z}\times\mathcal{Z}_{n}$.
Our main task is to determine which fissions of these $\mathfrak{S}$-subsets
are possible based upon assumptions on the $\mathfrak{S}$-subgroups. The
discrete case is always inherently easier to consider. For example, if we have
the discrete case and $\mathfrak{S}$ contains a singleton other than the
identity $\\{1\\}$, say $\\{a^{k}z^{t}\\}$, then by Lemma 2.2,
$(a^{k}z^{t})^{s}C$ is a primitive set of $\mathfrak{S}$ for all
$C\in\mathcal{D}(T(\mathfrak{S}))$ and all integers $s$. Hence, we have
$\mathfrak{S}_{\mathcal{Z}^{(t)}\times\mathcal{Z}_{n}}=F[\langle
a^{k}a^{t}\rangle]\times T(\mathfrak{S})\cong F[\mathcal{Z}]\times
T(\mathfrak{S})$ under this assumption. Therefore, if $z^{t}\mathcal{Z}_{n}$
contains a primitive subset which is a singleton then
$\mathfrak{S}_{\mathcal{Z}^{(t)}\times\mathcal{Z}_{n}}$ is traditional. When
$T(\mathfrak{S})$ is automorphic (which is always the case when $n$ is prime),
we may also conclude that
$\mathfrak{S}_{\mathcal{Z}^{(t)}\times\mathcal{Z}_{n}}$ automorphic. We will
use this fact about singletons frequently throughout the sequel.

######  Theorem 3.1.

Let $G=\mathcal{Z}\times\mathcal{Z}_{n}$ for some positive integer $n$. Let
$\mathfrak{S}$ be a Schur ring over $G$. Suppose $\mathcal{Z}^{(n)}$ is an
$\mathfrak{S}$-subgroup, and let $C$ be an $S$-subset. Then
$C\in\mathcal{D}(\mathfrak{S})$ if and only if
$C^{(k)}\in\mathcal{D}(\mathfrak{S})$ for all $k$ such that $\gcd(k,n)=1$.

###### Proof.

First, suppose that $C^{(k)}\in\mathcal{D}(\mathfrak{S})$. As $\gcd(k,n)=1$,
we have that $|C|=|C^{(k)}|$. If $C$ is imprimitive, then there exists some
proper primitive subset $D\subsetneq C$. Then $D^{(k)}\subsetneq C^{(k)}$,
which is a proper $\mathfrak{S}$-subset of a primitive set, which is a
contradiction. Thus, $C\in\mathcal{D}(\mathfrak{S})$.

Next, suppose that $C\in\mathcal{D}(\mathfrak{S})$. Assume first the discrete
case, that is, $\varphi(\mathfrak{S})=F[\mathcal{Z}]$. Then there is an
integer $t$ such that

$C=\\{a^{n_{1}}z^{t},a^{n_{2}}z^{t},\ldots,a^{n_{l}}z^{t}\\}\subseteq
z^{t}\mathcal{Z}_{n}.$

Then consider

$C^{(k)}=\\{a^{kn_{1}}z^{kt},a^{kn_{2}}z^{kt},\ldots,a^{kn_{l}}z^{kt}\\}\subseteq
z^{tk}\mathcal{Z}_{n},$

which is an $\mathfrak{S}$-subset. Suppose that $C^{(k)}$ is not primitive.
Then a subset of it must be, say $D$. Up to relabeling, we may assume
$D=\\{a^{kn_{1}}z^{kt},a^{kn_{2}}z^{kt},\ldots a^{kn_{h}}z^{kt}\\}$. As
$\mathcal{Z}^{(n)}$ is an $\mathfrak{S}$-subgroup, we know $\\{z^{n}\\}$ is
primitive. Also, there exist integers $r,s$ such that $kr+ns=1$. Now consider
the $\mathfrak{S}$-subset

$z^{nst}D^{(r)}=\\{a^{kn_{1}r}z^{ktr+nst},a^{kn_{2}r}z^{ktr+nst},\ldots,a^{kn_{h}r}z^{ktr+nst}\\}\\\
=\\{a^{n_{1}}z^{t},a^{n_{2}}z^{t},\ldots a^{n_{h}}z^{t}\\}.$

This is a strict $\mathfrak{S}$-subset of $C$, contradicting it being
primitive. Hence, $C^{(k)}$ must be primitive.

The case where $\varphi(\mathfrak{S})=F[\mathcal{Z}]^{\pm}$ is handled
similarly, where the $\mathfrak{S}$-subset $z^{nst}D^{(r)}$ is replaced with
$(\\{z^{nst},z^{-nst}\\}D^{(r)})\cap(\\{z^{t},z^{-t}\\}\mathcal{Z}_{n})$. ∎

######  Theorem 3.2.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{n}$ for some positive integer $n$.
Suppose $\mathfrak{S}$ is a Schur ring over $G$. Let $H$ be the maximal such
$\mathfrak{S}$-subgroup such that $H\leq\mathcal{Z}$. If
$n\mid[\mathcal{Z}:H]$, then let $K$ be the unique subgroup777In the case that
$[\mathcal{Z}:H]=\infty$, that is, $H=1$, set $K:=1$ and the above result and
proof would remain valid. of $\mathcal{Z}$ such that $[K:H]=n$. Then every
primitive set of $\mathfrak{S}$ contained in
$G\smallsetminus(K\times\mathcal{Z}_{n})$ is a union of cosets of some
nontrivial $\mathfrak{S}$-subgroup of $\mathcal{Z}_{n}$.

###### Proof.

First, suppose $\varphi(\mathfrak{S})=F[\mathcal{Z}].$ For a fixed integer
$t$, let $C\subseteq z^{t}\mathcal{Z}_{n}$ be an $\mathfrak{S}$-class. Then
there exists some subset $A\subseteq\mathcal{Z}_{n}$ such that $C=z^{t}A$.
Then consider the product
$\mathcal{X}:=\overline{C}\,\overline{C}^{(n-1)}=z^{nt}\overline{A}\,\overline{A}^{*}\in\mathfrak{S}.$
Clearly, the identity $1\in AA^{*}$ is a tycoon. If this pair has no other
tycoons, then $z^{nt}$ is the only element in $\mathcal{X}$ with
$|A|$-multiplicity, which would imply that $\\{z^{nt}\\}$ is an
$\mathfrak{S}$-subset and $\langle z^{nt}\rangle$ is an
$\mathfrak{S}$-subgroup. If $t$ is chosen such that
$z^{t}\in\mathcal{Z}\smallsetminus K$, then
$z^{nt}\in\mathcal{Z}\smallsetminus H$ and we have contradicted the maximality
of $H$ (since $\langle H,z^{nt}\rangle$ would be an $\mathfrak{S}$-subgroup).
Therefore, $(A,A)$ must have a second tycoon, which by Lemma 2.3 finishes this
case.

The case that $\varphi(\mathfrak{S})=F[\mathcal{Z}]^{\pm}$ is handled
similarly. Let $C\subseteq\\{z^{t},z^{-t}\\}\mathcal{Z}_{n}$ be a primitive
set, and let $C=z^{t}A\cup z^{-t}B$ for $A,B\subseteq\mathcal{Z}_{n}$.
Necessarily, $|A|=|B|$ (otherwise, $\varphi(S)$ would be discrete). Consider
the product

$\mathcal{X}:=(\overline{C}\,\overline{C}^{(n-1)})\circ\overline{\\{z^{nt},z^{-nt}\\}\mathcal{Z}_{n}}=z^{nt}\overline{A}\,\overline{A}^{*}+z^{-nt}\overline{B}\,\overline{B}^{*}\in\mathfrak{S}.$

Clearly, $1$ is a tycoon in $(A,A)$ and $(B,B)$. Then we have that $z^{nt}$
and $z^{-nt}$ have the same multiplicity of $|A|$ in $\mathcal{X}$. If these
are the only tycoons in the pairs $(A,A)$ and $(B,B)$, then
$\\{z^{nt},z^{-nt}\\}$ is primitive in $\mathfrak{S}$. If again $t$ is chosen
such that $z^{t}\in\mathcal{Z}\smallsetminus K$, then the maximality of $H$ is
contradicted. We, therefore, conclude that $(A,A)$ or $(B,B)$ must have a
second tycoon. In fact, they both do. To see this, the subset of
$\operatorname{supp}(\mathcal{X})$ consisting of all the same fixed
multiplicity is clearly an $\mathfrak{S}$-subset. In particular, the subset of
$\operatorname{supp}(\mathcal{X})$ with elements have multiplicity $|A|$ is an
$\mathfrak{S}$-subset, call it $D$, but $D$ is exactly the set of the tycoons
of $(A,A)$ and $(B,B)$. As $\varphi(D)$ is a symmetric set in
$\varphi(\mathfrak{S})$, the number of elements in $D$ of the form
$a^{i}z^{nt}$ must equal the number of elements in $D$ of the form
$a^{j}z^{-nt}$. In particular, the number of tycoons in $(A,A)$ is equal to
the number in $(B,B)$. By Lemma 2.3, $A$ and $B$ are unions of cosets of some
$\mathfrak{S}$-subgroup of $\mathcal{Z}_{n}$. Let $\mathcal{Z}_{d}$ be the
$\mathfrak{S}$-subgroup of $\mathcal{Z}_{n}$ which stabilizes $A$. Then

$\overline{C}\,\overline{\mathcal{Z}_{d}}=(z^{t}\overline{A}+z^{-t}\overline{B})\overline{\mathcal{Z}_{d}}=|d|z^{t}\overline{A}+z^{-t}\overline{B}\,\overline{\mathcal{Z}_{d}}=|d|\overline{C},$

where the last equality follows from the primitivity of $C$. Hence,
$B\mathcal{Z}_{d}=B$, and $C$ is a union of cosets of $\mathcal{Z}_{d}$. ∎

Let $p$ be a prime number. In the case $G=\mathcal{Z}\times\mathcal{Z}_{q}$,
where $q=p^{n}$, there is a unique minimal torsion $\mathfrak{S}$-subgroup in
$\mathcal{Z}_{q}$. Thus, every $\mathfrak{S}$-class outside of
$K\times\mathcal{Z}_{q}$ is a union of cosets of this unique minimal
$\mathfrak{S}$-subgroup. Hence, $\mathfrak{S}$ is necessarily a wedge product.
Furthermore, the above proof can easily be modified with the product
$\overline{C}\,\overline{C}^{(p-1)}$, which gives a tighter control on the
structure of primitive sets in $\mathfrak{S}$. We summarize this in the
following corollary.

######  Corollary 3.3.

Let $G=\mathcal{Z}\times\mathcal{Z}_{q}$, where $q=p^{n}$ for some prime $p$.
Let $\mathfrak{S}$ be a Schur ring over $G$. Let $H$ be the maximal
$\mathfrak{S}$-subgroup such that $H\leq\mathcal{Z}$. If
$p\mid[\mathcal{Z}:H]$, then let $K$ be the unique subgroup of $\mathcal{Z}$
such that $[K:H]=p$. Then $\mathfrak{S}$ is a wedge product. In particular,
$\mathfrak{S}=\mathfrak{S}_{K\times\mathcal{Z}_{q}}\wedge\varphi(\mathfrak{S})$.

We next consider the case that $G=\mathcal{Z}\times\mathcal{Z}_{p}$, for some
prime $p$. Note that the classification of Schur rings over cyclic groups
simplifies when the cyclic group has prime order. As direct and wedge products
are impossible over $\mathcal{Z}_{p}$ and the trivial Schur ring is
automorphic, all Schur rings over $\mathcal{Z}_{p}$ are automorphic and
correspond to an integer $m$ coprime to the order $p$.

Before continuing, we stop to discuss
$\operatorname*{Aut}(\mathcal{Z}\times\mathcal{Z}_{p})$. Let $\sigma_{m}:G\to
G$ be the automorphism associated with the rule $\sigma_{m}(z)=z$, and
$\sigma_{m}(a)=a^{m}$ for each integer $m$ coprime to $p$. Also, let
$\rho:G\to G$ be the automorphism defined by $\rho(z)=az$ and $\rho(a)=a$.
Finally, let $r$ be a primitive root modulo $p$. Then
$\operatorname*{Aut}(\mathcal{Z}\times\mathcal{Z}_{p})=\langle\rho,\sigma_{r},*\rangle\cong
GA(1,p)\times\mathcal{Z}_{2}$, where $GA(1,p)$ denotes the general affine
group over a finite vector space of order $p$.

######  Theorem 3.4.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$, where $p$ is a prime. If
$\mathfrak{S}$ is a Schur ring over $G$ where $\mathcal{Z}$ is an
$\mathfrak{S}$-subgroup, then $\mathfrak{S}$ is an automorphic Schur ring.

###### Proof.

First, suppose $\varphi(\mathfrak{S})=F[\mathcal{Z}]$, which implies that
$\mathfrak{S}_{\mathcal{Z}}=F[G]$, that is, $\\{z^{k}\\}$ is a primitive set
for each integer $k$. If $C\in\mathcal{D}(T(\mathfrak{S}))$, then $z^{k}C$ is
likewise primitive in $\mathfrak{S}$ by Lemma 2.2. Hence,
$\mathfrak{S}=F[G]\times T(\mathfrak{S})$.888Note that this argument does not
require $p$ to be prime and hence applies to all positive integers $n$. As
$\mathfrak{S}_{\mathcal{Z}}$ and $\mathfrak{S}_{p}$ are automorphic,
$\mathfrak{S}$ is automorphic as well.

Next, suppose $\varphi(\mathfrak{S})$ is symmetric, which implies that
$\mathfrak{S}_{\mathcal{Z}}=F[\mathcal{Z}]^{\pm}$, that is,
$\\{z^{k},z^{-k}\\}$ is a primitive set for each integer $k$. Then if
$C\in\mathcal{D}(T(\mathfrak{S}))$, then $\\{z^{k},z^{-k}\\}C$ is an
$\mathfrak{S}$-subset for every $k$. If all $\mathfrak{S}$-subsets of this
form are primitive for every $C$ and every $k$, then again
$\mathfrak{S}=F[G]^{\pm}\times T(\mathfrak{S})$ is automorphic.

Suppose then that there is some integer $k$ and some primitive set
$C\in\mathcal{D}(T(\mathfrak{S}))$ such that $\\{z^{k},z^{-k}\\}C$ is
imprimitive and that $D\subsetneq\\{z^{k},z^{-k}\\}C$ is a primitive subset.
Since $\varphi(D)$ is symmetric, $D=z^{k}A\cup z^{-k}B$ for $A,B\subseteq C$
and $|A|=|B|$. Since all non-identity elements of $\mathcal{Z}_{p}$ are
generators, we may assume that $a\in A$. If $\pi:G\to\mathcal{Z}_{p}$ is
natural projection onto $\mathcal{Z}_{p}$, then
$\pi(\overline{D})=\pi(\overline{A}+\overline{B})=\overline{A}+\overline{B}$
and $\pi(\overline{C})=\overline{C}$. But
$\operatorname{supp}(\overline{A}+\overline{B})=A\cup B\subseteq
C\in\mathcal{D}(T(\mathfrak{S}))$. Hence, $A\cup B=C$ and $A\cap B=\emptyset$.

Let $m$ be the integer coprime to $p$ which affords the automorphic Schur ring
$T(\mathfrak{S})$ over $\mathcal{Z}_{p}$, which necessarily has even order. We
claim

$D=z^{k}\\{a^{m^{2t}}\mid t\in\mathcal{Z}\\}\ \cup\ z^{-k}\\{a^{m^{2t+1}}\mid
t\in\mathcal{Z}\\}=\\{az^{k},a^{m}z^{-k},a^{m^{2}}z^{k},a^{m^{3}}z^{-k},\ldots\\}.$

If the claim were false, then, up to relabeling of generators, we may assume
$D=\\{az^{k},a^{m}z^{k},a^{m^{2}}z^{-k},\ldots\\}$. Next,

$D^{(m)}=z^{km}A^{(m)}\cup
z^{-km}B^{(m)}=\\{a^{m}z^{km},a^{m^{2}}z^{km},a^{m^{3}}z^{-km},\ldots\\}$

and

$\\{z^{k(m-1)},z^{k(1-m)}\\}D=z^{km}A\cup z^{k(2-m)}A\cup z^{k(m-2)}B\cup
z^{-km}B=\\\
\\{az^{km},a^{m}z^{km},a^{m^{2}}z^{k(m-2)},\ldots,az^{k(2-m)},a^{m}z^{k(2-m)},a^{m^{2}}z^{-km},\ldots\\},$

which are both $\mathfrak{S}$-subsets. By Theorem 3.1, $D^{(m)}$ is primitive.
As $a^{m}z^{km}\in D^{(m)}\cap\\{z^{k(m-1)},z^{k(1-m)}\\}D$, we have that
$D^{(m)}\subseteq\\{z^{k(m-1)},z^{k(1-m)}\\}D$. Then as $a^{m^{2}}z^{km}\in
D^{(m)}$ we have that $a^{m^{2}}z^{km}\in\\{z^{k(m-1)},z^{k(1-m)}\\}D$. But
this implies that $a^{m^{2}}z^{k}\in D$ and $a^{m^{2}}\in A\cap
B\neq\emptyset$, which is a contradiction. This proves the claim. Let
$D^{\prime}=z^{k}B\cup z^{-k}A$. Hence, $D\cup D^{\prime}=\\{z^{k},z^{-k}\\}C$
and $D,D^{\prime}\in\mathcal{D}(\mathfrak{S})$.

If $\ell$ is some other integer, note that
$\\{z^{\ell-k},z^{k-\ell}\\}D\cap\\{z^{\ell},z^{-\ell}\\}C$ is a non-empty
$\mathfrak{S}$-subset, as they both contain $az^{\ell}$, but it is a proper
subset of $\\{z^{\ell},z^{-\ell}\\}C$ since it does not contain $az^{-\ell}$.
Applying the above argument, $\\{z^{\ell},z^{-\ell}\\}C=E\cup E^{\prime}$,
where $E,E^{\prime}\in\mathcal{D}(\mathfrak{S})$ and $|E|=|E^{\prime}|$.

Finally, if $C^{\prime}\in\mathcal{D}(T(\mathfrak{S}))$ is some other non-
identity class, then $C^{\prime}=C^{(s)}$ for some integer $r$ coprime to $p$.
Let $\\{z,z^{-1}\\}C=E\cup E^{\prime}$ be a primitive decomposition. Then
$E^{(s)}=E^{\prime(s)}=\\{z^{s},z^{-s}\\}C^{\prime}$ and
$E^{(r)},E^{\prime(s)}\in\mathcal{D}(\mathfrak{S})$. Hence,
$\\{z^{s},z^{-s}\\}C^{\prime}$ is imprimitive, which implies that
$\\{z^{\ell},z^{-\ell}\\}C^{\prime}$ is imprimitive with an analogous
primitive decomposition as illustrated above. In summary, each primitive set
of $\mathfrak{S}$ has the form

$\\{a^{i}z^{k},a^{im}z^{-k},a^{im^{2}}z^{k},a^{im^{3}}z^{-k},\ldots\\}$

for some $i$. But these are exactly the orbits of the automorphism
$\sigma_{-m}*$. Therefore, $\mathfrak{S}=F[G]^{\langle\sigma_{-m}*\rangle}$. ∎

######  Corollary 3.5.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$, where $p$ is a prime. If
$\mathfrak{S}$ is a Schur ring over $G$ where $\langle a^{i}z\rangle$ is an
$\mathfrak{S}$-subgroup for any integer $i$, then $\mathfrak{S}$ is an
automorphic Schur ring.

###### Proof.

The image $\rho^{-i}(\mathfrak{S})$ is a Schur ring over $G$ such that
$\rho^{-i}(\langle a^{i}z\rangle)=\langle z\rangle=\mathcal{Z}$ is an
$\rho^{-i}(\mathfrak{S})$-subgroup. By Theorem 3.4, $\rho^{-i}(\mathfrak{S})$
is automorphic. As the automorphic image of an automorphic Schur ring is
likewise automorphic,999If $\mathfrak{S}=F[G]^{\mathcal{H}}$ and
$\tau\in\operatorname*{Aut}(G)$, then
$\tau(\mathfrak{S})=F[G]^{\tau\mathcal{H}\tau^{-1}}$.
$\mathfrak{S}=\rho^{i}(\rho^{-i}(\mathfrak{S}))$ is likewise automorphic. ∎

######  Corollary 3.6.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$ where p is a prime.101010In the
proof of Theorem 3.10, we will assume that $p$ is an odd prime. The case when
$p=2$ is fairly simple and is already addressed in [1]. Let $\mathfrak{S}$ be
a Schur ring over $G$ such that $\varphi(\mathfrak{S})=F[\mathcal{Z}]^{\pm}$.
If $C\in\mathcal{D}(\mathfrak{S})$ such that $|C|=2$ and $C\subseteq
z\mathcal{Z}_{p}\cup z^{-1}\mathcal{Z}_{p}$, then $\mathfrak{S}$ is an
automorphic Schur ring.

###### Proof.

Suppose $C=\\{a^{k}z,a^{\ell}z^{-1}\\}$ is primitive for some $k$ and $\ell$.
Then consider the product
$\overline{C}^{2}=a^{2k}z^{2}+a^{2\ell}z^{-2}+2a^{k+\ell}\in\mathfrak{S}$. If
$a^{k+\ell}$ is the identity, then $\ell\equiv-k\pmod{p}$. In this case
$\langle C\rangle=\langle a^{k}z\rangle\cong\mathcal{Z}$, which is an
$\mathfrak{S}$-subgroup. Then, by Corollary 3.5, $\mathfrak{S}$ is an
automorphic Schur ring. If $a^{k+\ell}\neq 1$, then this means that
$T(\mathfrak{S})$ is discrete, as the primitive set $\\{a^{k+\ell}\\}$
generates $\mathcal{Z}_{p}$. Then
$a^{-(k+\ell)/2}C=\\{a^{(k-\ell)/2}z,a^{-(k-\ell)/2}z^{-1}\\}$, which is
primitive by Lemma 2.2. Then $\langle
a^{-(k+\ell)/2}C\rangle\cong\mathcal{Z}$. Again, we have that $\mathfrak{S}$
is an automorphic Schur ring. ∎

######  Theorem 3.7.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$, where $p$ is a prime. Let
$\mathfrak{S}$ be a Schur ring over $G$ such that $H$ is the maximal
$\mathfrak{S}$-subgroup contained in $\mathcal{Z}$. Let
$T(\mathfrak{S})=F[\mathcal{Z}_{p}]^{\mathcal{H}}$ be an automorphic Schur
ring with $|\mathcal{H}|=m$. If there is a primitive set
$C\in\mathcal{D}(\mathfrak{S})$ such that $C\subseteq\mathcal{Z}\smallsetminus
H$ and $\gcd(|C|,m)=1$, then $\mathfrak{S}$ is a wedge product or an
automorphic Schur ring.

###### Proof.

Because of Corollary 3.3 and Theorem 3.4, it suffices to prove the case where
$H=\mathcal{Z}^{(p)}$. First, suppose $\varphi(\mathfrak{S})=F[\mathcal{Z}]$.
Suppose that $\gcd(|C|,m)=1$ for some primitive set $C=z^{t}A$, for some
$A\subseteq\mathcal{Z}_{p}$. By Theorem 3.1, we may suppose $t=1$. If $|A|=1$,
then $\mathfrak{S}$ is an automorphic Schur ring by previous reasoning.
Suppose then that $|A|>1$. Without the loss of generality, we may suppose that
$a,a^{k}\in A$ are distinct elements. Take $D\in\mathcal{D}(T(\mathfrak{S}))$
such that $a^{k-1}\in D$. By Lemma 2.2, since $|D|=m$, we know that
$\overline{C}\,\overline{D}=\lambda\overline{E}$, where
$E\in\mathcal{D}(\mathfrak{S})$. Necessarily, $E\subseteq z\mathcal{Z}_{p}$
and $(az)a^{k-1}=a^{k}z\in CD=E$. As $a^{k}z\in C\cap E$ is contained in two
primitive sets, we conclude that $C=E$, that is, $CD=C$. So, it must be that
$D\subseteq G_{C}$, which implies $G_{C}=\mathcal{Z}_{p}$. In particular,
$A=\mathcal{Z}_{p}$, that is, $C=z\mathcal{Z}_{p}$. Hence,
$\mathfrak{S}=\mathfrak{S}_{H}\wedge F[\mathcal{Z}]$.

The symmetric case is handled similarly. If
$\varphi(\mathfrak{S})=F[\mathcal{Z}]^{\pm}$ and $\gcd(|C|,m)=1$ for some
primitive set $C=zA\cup z^{-1}B$, for some $A,B\subseteq\mathcal{Z}_{p}$, then
the case where $|A|=1$ is handled by Corollary 3.6 as $|C|=2$ and the case
where $|A|>1$ is handled just like the discrete case, that is, we conclude
that $A=B=\mathcal{Z}_{p}$ by considering stabilizers. Thus, $\mathfrak{S}$ is
either automorphic or wedge decomposable. ∎

######  Corollary 3.8.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$ where $p$ is a prime. Let
$\mathfrak{S}$ be a Schur ring over $G$ such that $\mathcal{Z}^{(p)}$ is an
$\mathfrak{S}$-subgroup. Let
$T(\mathfrak{S})=F[\mathcal{Z}_{p}]^{\mathcal{H}}$ be an automorphic Schur
ring with $|\mathcal{H}|=m$. If there is a primitive set
$C\in\mathcal{D}(\mathfrak{S})$ such that
$C\subseteq\mathcal{Z}\smallsetminus\mathcal{Z}^{(p)}$ and $\gcd(|C|,m)=1$,
then $\mathfrak{S}$ is an automorphic Schur ring.

###### Proof.

If $\langle a^{i}z\rangle$ is an $\mathfrak{S}$-subgroup for some $i$, then
$\mathfrak{S}$ is automorphic by Corollary 3.5. So, we may assume that
$\mathcal{Z}^{(p)}$ is a maximal $\mathfrak{S}$-subgroup. By Theorem 3.7,
$\mathfrak{S}$ is either automorphic or a wedge product of the form
$F[\mathcal{Z}]\wedge F[\mathcal{Z}]$ or $F[\mathcal{Z}]^{\pm}\wedge
F[\mathcal{Z}]^{\pm}$, with decomposition $1<H\times\mathcal{Z}_{p}\leq G$. In
the latter two cases, these wedge products are equal to
$F[G]^{\langle\rho\rangle}$ and $F[G]^{\langle\rho,*\rangle}$, respectively. ∎

######  Theorem 3.9.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$ where $p$ is a prime. Let
$\mathfrak{S}$ be a Schur ring over $G$ such that $\mathcal{Z}^{(p)}$ is an
$\mathfrak{S}$-subgroup. Let
$T(\mathfrak{S})=F[\mathcal{Z}_{p}]^{\mathcal{H}}$ be an automorphic Schur
ring with $|\mathcal{H}|=m$. If $m$ is a power of a prime, then $\mathfrak{S}$
is an automorphic Schur ring.

###### Proof.

Let $m=q^{k}$, where $q$ is prime. As $|z\mathcal{Z}_{p}|=p$, there is some
primitive subset $C$ of $z\mathcal{Z}_{p}$ such that $q\nmid|C|$. Then
Corollary 3.8 applies.

Consider the symmetric case then. Note that
$|\\{z,z^{-1}\\}\mathcal{Z}_{p}|=2p$, so if $q$ is an odd prime, then there is
some primitive subset $C$ of $\\{z,z^{-1}\\}\mathcal{Z}_{p}$ such that
$q\nmid|C|$. Then Corollary 3.8 applies again.

Suppose $m=2^{\ell}$ for some $\ell\geq 1$. We may choose the primitive set
$C=zA\cup z^{-1}B$ so that $A$ has the smallest possible odd length (such a
choice exists as the union of all possible sets $A$ is $\mathcal{Z}_{p}$ and
not all lengths could be even). Thus, $\gcd(|A|,m)=1$. If $|A|=1$, then
$\mathfrak{S}$ would be automorphic by Corollary 3.6. So, without the loss of
generality, we may assume that $a,a^{k}\in A$. Let
$D\in\mathcal{D}(T(\mathfrak{S}))$ such that $a^{k-1}\in D$. Thus,
$\overline{C}\,\overline{D}=\lambda\overline{C}+\cdots$. Then
$\overline{C}^{*}\overline{C}=\mu\overline{D}+\cdots$ where
$\lambda|C|=\mu|D|$ by Lemma 2.1. So, $2\lambda|A|=\mu m$, which implies that
$m\mid 2\lambda$. Hence, $\lambda=2^{\ell-1}t$ for some nonzero integer $t$.
Now, as $1\leq\lambda\leq\min(|C|,|D|)=\min(2|A|,2^{\ell})$, it must be that
$t=1$ or $t=2$.

If $t=2$, then it must be that
$\overline{C}\,\overline{D}=2^{\ell}\overline{C}$ as both sides of the
equality count the same number of elements. Since $D$ necessarily stabilizes
$A$, we see again that $A=\mathcal{Z}_{p}$, that is,
$C=\\{z,z^{-1}\\}\mathcal{Z}_{p}$. By Theorem 3.1, the primitive sets
belonging to $G\smallsetminus(\mathcal{Z}^{(p)}\times\mathcal{Z}_{p})$ all
have the form $\\{z^{k},z^{-k}\\}\mathcal{Z}_{p}$ for some integer $k$. As
$\mathfrak{S}_{\mathcal{Z}^{(p)}\times\mathcal{Z}_{p}}=F[\mathcal{Z}^{(p)}]\times
F[\mathcal{Z}_{p}]^{\mathcal{H}}$, we have
$\mathfrak{S}=F[G]^{\langle\rho,\sigma_{r},*\rangle}$, where $|\sigma_{r}|=m$
in $\operatorname*{Aut}(\mathcal{Z}_{p})$.

In the above argument, the choice of $t=1$ or $t=2$ depends upon the choice of
the primitive set $D$. This primitive set could have been interchanged for any
other primitive set $D^{\prime}$ such that $a^{u-v}\in D^{\prime}$ and
$a^{u},a^{v}\in A$. Let $\mathcal{D}:=\\{D\in\mathcal{D}(T(\mathfrak{S}))\mid
xy^{-1}\in D\text{ for some }x,y\in C\\}$. For any set $D\in\mathcal{D}$, if
$t=2$, then the above argument applies and we are done. Suppose then that
$t=1$ for all $D\in\mathcal{D}$. Then
$\overline{C}\,\overline{D}=\frac{m}{2}\overline{C}+\cdots$, where the right-
hand side so far only accounts for half of the elements in the product $CD$.
On the other hand,

$\overline{C}^{*}\overline{C}=2|A|+\sum_{D\in\mathcal{D}}\mu_{D}D,$

where $\mu_{D}|D|=\frac{m}{2}|C|$ by Lemma 2.1. Hence,
$\mu_{D}=\dfrac{m(2|A|)}{2m}=|A|$. Counting multiplicities in the above
equation, the left-hand side contains $|C|^{2}=4|A|^{2}$ many elements, and
the right-hand side contains $2|A|+|\mathcal{D}||A|m$ many elements. Hence,
$4|A|^{2}=2|A|+|\mathcal{D}||A|m$, or, simply,
$2|A|=1+2^{\ell-1}|\mathcal{D}|$. As the left-hand side is clearly even, the
right-hand side must also be even, which implies that $\ell=1$, that is, $m=2$
and $\mathcal{H}=\langle*\rangle$. This observation simplifies the equation
involving $\overline{C}\,\overline{D}$ to
$\overline{C}(a^{k-1}+a^{1-k})=\overline{C}+\overline{C_{1}}$, where
$C_{1}\in\mathcal{D}(\mathfrak{S})$ and $|C_{1}|=|C|$. This follows from the
minimality of the choice of $A$. We note that $a^{2-k}z=az(a^{1-k})\in
C\\{a^{k-1},a^{1-k}\\}$. If $a^{2-k}\in A$, then
$az=a^{k}z(a^{1-k})=a^{2-k}z(a^{k-1})$ appears with multiplicity 2 in
$C\\{a^{k-1},a^{1-k}\\}$. This implies
$\overline{C}(a^{k-1}+a^{1-k})=2\overline{C}$, a contradiction. Thus,
$a^{2-k}z\in C_{1}$.

Now, consider $\overline{C_{1}}(a^{k-1}+a^{1-k})$. As $a^{2-k}z\in C_{1}$, we
know $az\in C_{1}\\{a^{k-1},a^{1-k}\\}$. Thus,
$\overline{C_{1}}(a^{k-1}+a^{k-1})=\overline{C}+\overline{C_{2}}$, where
$C_{2}\in\mathcal{D}(\mathfrak{S})$ and $|C|=|C_{2}|$, by similar reasoning as
before. If $C_{2}=C_{1}$, then $\\{a^{k-1},a^{1-k}\\}$ stabilizes $C\cup
C_{1}$. This implies that $C\cup C_{1}=\\{z,z^{-1}\\}\mathcal{Z}_{p}$, but
this implies that $|A|$ divides $p$. Then $|A|=1$ or $|A|=p$. Both of these
cases imply $\mathfrak{S}$ is automorphic.

Suppose then that $C_{2}\neq C_{1}$. As $a^{3-2k}z=a^{2-k}z(a^{1-k})\in
C_{1}\\{a^{k-1},a^{1-k}\\}$, $a^{3-2k}z\in C$ or $a^{3-2k}z\in C_{2}$. In the
former case, $a^{2-k}z=az(a^{1-k})=a^{3-2k}z(a^{k-1})$ appears with
multiplicity 2 in $C\\{a^{k-1},a^{1-k}\\}$, which again implies that
$C=C_{1}$. Thus, it must be that $a^{3-2k}z\in C_{2}$, and this implies that
the process continues.

Let $C_{0}=C$. Eventually, there will exist some integer $n$ where
$C_{n}=C_{i}$ for some $i<n$. Then
$\left(\sum_{j=0}^{n}\overline{C}_{j}\right)(a^{k-1}+a^{1-k})=\sum_{j=0}^{n}\mu_{j}\overline{C}_{j}$,
which shows that $\\{a^{k-1},a^{1-k}\\}$ stabilizes $\bigcup_{j=0}^{n}C_{j}$.
This again implies that $|A|=1$ or $|A|=p$. Thus, $\mathfrak{S}$ is
automorphic if $m=2^{\ell}$. ∎

######  Theorem 3.10.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$ where $p$ is a prime.111111In the
proof of Theorem 3.10, we will assume that $p$ is an odd prime. The case when
$p=2$ is fairly simple and is already addressed in [1]. Let $\mathfrak{S}$ be
a Schur ring over $G$ such that $\mathcal{Z}^{(p)}$ is an
$\mathfrak{S}$-subgroup. Suppose $z\mathcal{Z}_{p}$ or
$\\{z,z^{-1}\\}\mathcal{Z}_{p}$ is the union of exactly two primitive sets.
Then $\mathfrak{S}$ is an automorphic Schur ring.

###### Proof.

First, consider the discrete case. If $\\{z,z^{-1}\\}\mathcal{Z}_{p}$ is a
union of two primitive sets, then they must necessarily be $z\mathcal{Z}_{p}$
and $z^{-1}\mathcal{Z}_{p}$. Thus,
$\mathfrak{S}=F[G]^{\langle\rho,\sigma_{m}\rangle}$. Consider then the case
where $z\mathcal{Z}_{p}=C\cup D$ for primitive sets
$C,D\in\mathcal{D}(\mathfrak{S})$. As $p$ is a prime, it must be that
$\gcd(|C|,|D|)=1$. Let $D$ be chosen so that $|D|>|C|$. By Lemma 2.2, we know
that $\overline{C}\,\overline{D}=\lambda\overline{E}$, for some
$E\in\mathcal{D}(\mathfrak{S})$ and $E\subseteq z^{2}\mathcal{Z}_{p}$. By
Theorem 3.1, we know that the primitive sets forming $z^{2}\mathcal{Z}$ are
$C^{(2)}$ and $D^{(2)}$. If $E=C^{(2)}$, then $|E|=|C|$ and $\lambda=|D|$, but
this implies $\lambda>|C|$, which is a contradiction as the multiplicity on
each term is bounded by $\min(|C|,|D|)$. If $E=D^{(2)}$, then $|E|=|D|$ and
$\lambda=|C|$. Say that $C=zA$ for some $A\subseteq\mathcal{Z}_{p}$. Then

$\overline{C}\,\overline{C}+\overline{C}\,\overline{D}=\overline{C}\,(z\overline{\mathcal{Z}_{p}})=z^{2}(\overline{A}\,\overline{\mathcal{Z}_{p}})=\lambda(z^{2}\overline{\mathcal{Z}_{p}})=\lambda\overline{C^{(2)}}+\lambda\overline{D^{(2)}}.$

Since $\overline{C}\,\overline{D}=\lambda\overline{D^{(2)}}$, we have that
$\overline{C}\,\overline{C}=\lambda\overline{C^{(2)}}$, that is,
$\overline{C}^{2}=\lambda\overline{C^{(2)}}$. Thus,
$\overline{A}\,(\overline{A^{*}})^{*}=\overline{A}\,\overline{A}=\lambda\overline{A^{(2)}}$.
As every element in $A^{2}$ is a tycoon of $(A,A^{*})$ ($\lambda=|A|$), if
$\lambda\neq 1$ then $A=\mathcal{Z}_{p}$, which contradicts $\mathcal{Z}_{p}$
being a union of two primitive sets. Thus, $|A|=1=|C|$, and $\mathfrak{S}$ is
automorphic, as $C$ is a singleton.

Now consider the case where $\varphi(\mathfrak{S})=F[\mathcal{Z}]^{\pm}$. As
$z\mathcal{Z}_{p}$ is not an $\mathfrak{S}$-subset, we need only consider the
case $\\{z,z^{-1}\\}\mathcal{Z}_{p}=C\cup D$, where
$C,D\in\mathcal{D}(\mathfrak{S})$. Let $C=zA\cup z^{-1}B$ and
$D=zA^{\prime}\cup z^{-1}B^{\prime}$. Necessarily, $\gcd(|A|,|A^{\prime}|)=1$
as $p$ is prime. Choose $D$ such that $|A|<|A^{\prime}|$. Consider the product

$\overline{C}\,\overline{D}=(z^{2}\overline{A}\,\overline{A^{\prime}}+z^{-2}\overline{B}\,\overline{B^{\prime}})+(\overline{B}\,\overline{A^{\prime}}+\overline{A}\,\overline{B^{\prime}})$

Note that clearly $(z^{2}AA^{\prime}\cup
z^{-2}BB^{\prime})\subseteq\\{z^{2},z^{-2}\\}\mathcal{Z}_{p}$ and
$(BA^{\prime}\cup AB^{\prime})\subseteq{\mathcal{Z}_{p}}$. By Theorem 3.1, the
only primitive sets in $\\{z^{2},z^{-2}\\}\mathcal{Z}_{p}$ are $C^{(2)}$ and
$D^{(2)}$. This means that

$z^{2}\overline{A}\,\overline{A^{\prime}}+z^{-2}\overline{B}\,\overline{B^{\prime}}=\lambda\overline{C^{(2)}}+\mu\overline{D^{(2)}},$

for some integers $\lambda,\mu$. Hence,

$|A||A^{\prime}|=\lambda|A^{(2)}|+\mu|A^{\prime(2)}|=\lambda|A|+\mu|A^{\prime}|,$

which implies that $|A^{\prime}|\mid\lambda$. But $\lambda$ must be an integer
such that $0\leq\lambda\leq\min(|A|,|A^{\prime}|)=|A|$, which implies that
$\lambda=0$. Similarly, $\mu=|A|$ since $|A|\mid\mu$ and $0\leq\mu\leq|A|$
($\lambda$ and $\mu$ cannot both be $0$). Therefore,
$z^{2}\overline{A}\,\overline{A^{\prime}}+z^{-2}\overline{B}\,\overline{B^{\prime}}=\mu\overline{D^{(2)}}$,
that is,

$\overline{C}\,\overline{D}=\mu\overline{D}^{(2)}+\mathcal{X}$

for some $\mathcal{X}\in T(\mathfrak{S})$. Next consider

$\overline{C}\,\overline{C}+\overline{C}\,\overline{D}=\overline{C}(z+z^{-1})\overline{\mathcal{Z}_{p}}=\mu(z^{2}+z^{-2})\overline{\mathcal{Z}_{p}}+2\mu\overline{\mathcal{Z}_{p}}=\mu(\overline{C^{(2)}}+\overline{D^{(2)}})+2\mu\overline{\mathcal{Z}_{p}}.$

Considering those elements contained in
$(z^{2}+z^{-2})\overline{\mathcal{Z}_{p}}$ coming from
$\overline{C}\,\overline{C}$, we have that

$\overline{C}\,\overline{C}=\mu\overline{C^{(2)}}+\mathcal{Y},$

for some $\mathcal{Y}\in T(\mathfrak{S})$. On the other hand,

$\overline{C}\,\overline{C}=(z\overline{A}+z^{-1}\overline{B})^{2}=z^{2}\overline{A}^{2}+z^{-2}\overline{B}^{2}+2\overline{A}\,\overline{B}.$

Comparing terms, we have
$z^{2}\overline{A}^{2}+z^{-2}\overline{B}^{2}=\mu\overline{C^{(2)}}$. We also
know that
$\overline{C^{(2)}}=(z\overline{A}+z^{-1}\overline{B})^{(2)}=z^{2}\overline{A^{(2)}}+z^{-2}\overline{B^{(2)}}$.
So this means that $\overline{A}^{2}=\mu\overline{A^{(2)}}$. As $|A|=\mu$, we
see all the elements of $A^{2}$ are tycoons in $(A,A^{*})$. If $|A|>1$, then
we again have that $A=\mathcal{Z}_{p}$, which contradicts
$z\mathcal{Z}_{p}\cup z^{-1}\mathcal{Z}_{p}$ having two primitive subsets.
Thus, $|A|=1$, which implies that $\mathfrak{S}$ is automorphic by Corollary
3.6. ∎

##  4 Proof of Theorem 1.1

We summarize the results we have found thus far and how these provide the
proof of Theorem 1.1. In the special case that
$G=\mathcal{Z}\times\mathcal{Z}_{p}$ for some prime $p$, we have a strong
understanding of the primitive sets of any Schur rings $\mathfrak{S}$ over $G$
which we summarize here. Let $H$ be the maximal $\mathfrak{S}$-subgroup
contained in $\mathcal{Z}$. We have seen that if $H$ is trivial, then
$\mathfrak{S}$ is a wedge product by Corollary 3.3, which is traditional. When
$H$ is nontrivial, the Schur subring $\mathfrak{S}_{H\times\mathcal{Z}_{p}}$
is necessarily automorphic by Theorem 3.4. If there is a subgroup $K$ of
$\mathcal{Z}$ such that $[K:H]=p$, then $\mathfrak{S}$ is a wedge product. If
$\mathfrak{S}_{K\times\mathcal{Z}_{p}}$ is traditional, then so is
$\mathfrak{S}$. Therefore, we know the structure of all primitive sets except
those in
$\mathcal{D}(\mathfrak{S}_{K\times\mathcal{Z}_{p}})\smallsetminus\mathcal{D}(\mathfrak{S}_{H\times\mathcal{Z}_{p}})$.
This is why Theorems 3.7-3.10 focused on the case where $H=\mathcal{Z}^{(p)}$.

By Theorem 3.1, it suffices to determine the structure of the primitive
subsets of $z\mathcal{Z}_{p}$ or $\\{z,z^{-1}\\}\mathcal{Z}_{p}$ (depending on
the image $\varphi(\mathfrak{S})$). If $\mathfrak{S}$ contains the primitive
set $\\{a^{k}z\\}$ or $\\{a^{k}z,a^{-k}z^{-1}\\}$ for any integer $k$, then
Corollary 3.5 applies and $\mathfrak{S}$ is automorphic. If
$\\{z,z^{-1}\\}\mathcal{Z}_{p}$ has any primitive subset of length two, then
$\mathfrak{S}$ is automorphic by Corollary 3.6. Finally, since the torsion
subgroup $\mathcal{Z}_{p}$ is order $p$, we know that $T(\mathfrak{S})$ is an
automorphic Schur ring with all non-identity primitive sets having equal
length of $m$. The proofs of Theorems 3.4 and 3.7 utilized this fact. Theorem
4.1 will show in the case that $p$ is a Fermat or safe prime that
$\mathfrak{S}_{K\times\mathcal{Z}_{p}}$ is necessarily automorphic, which
imply that $\mathfrak{S}$ is traditional in this case. For general $p$, the
integer $m$ is necessarily a divisor of $p-1$. In the case of a Fermat or safe
prime, $p-1$ has very few divisors. This will complete the proof of Theorem
1.1.

######  Theorem 4.1.

Suppose $G=\mathcal{Z}\times\mathcal{Z}_{p}$ where $p$ is a Fermat or safe
prime. If $\mathfrak{S}$ is a Schur ring over $G$ where $\mathcal{Z}^{(p)}$ is
an $\mathfrak{S}$-subgroup, then $\mathfrak{S}$ is an automorphic Schur ring.

###### Proof.

Let $H=\mathcal{Z}^{(p)}$. We know that
$\mathfrak{S}_{H\times\mathcal{Z}_{p}}$ is an automorphic Schur ring by
Theorem 3.4. Let each non-identity primitive set in $T(\mathfrak{S})$ have
length $m\mid(p-1)$. If $p$ is a Fermat prime, then $m=2^{\ell}$ for some
$\ell$. If $p=2q+1$ is a safe prime, then $m=1$, $2$, $q$, or $2q$. Thus, we
will show that if $m=1$, $2^{\ell}$, $q$, or $2q$, then $\mathfrak{S}$ is
automorphic. By Theorem 3.9, we need only consider the case $m=2q$ (hence
$p=2q+1$ is a safe prime). Note in this case $T(\mathfrak{S})$ is trivial.

First consider the case where $\varphi(\mathfrak{S})=F[\mathcal{Z}]$. Let
$C=zA\subseteq z\mathcal{Z}_{p}$ be a primitive set. If $|A|=1$, then
$\mathfrak{S}$ is automorphic. So, we assume $|A|>1$. Note
$CC^{*}=AA^{*}\subseteq\mathcal{Z}_{p}$. Say $|A|=k$. By counting
multiplicities in $\overline{A}\,\overline{A}^{*}$ and considering that
$T(\mathfrak{S})$ is trivial, then there must be some positive integer
$\lambda$ such that $k(k-1)=\lambda(p-1)=2\lambda q$. In fact, this implies
that $A$ is a difference set of $\mathcal{Z}_{p}$. Since $q\mid k$ or
$q\mid(k-1)$, it must be that $k=0$, $1$, $q$, $q+1$, $2q$, or $2q+1$. As $A$
is non-empty and $z\mathcal{Z}_{p}$ does not contain a primitive subset which
is a singleton, we may rule out $k=0,1,2q$. If $k=2q+1=p$, then
$\mathfrak{S}=F[G]^{\langle\rho,\sigma_{r}\rangle}$, where $r$ is a primitive
root of $p$. If $k=q+1$, then $z\mathcal{Z}_{p}$ contains a primitive subset
whose length is either $1$ or $q$. Therefore, the only case that needs further
pursuit is $k=q$. If $k=q$, then the other primitive set in $z\mathcal{Z}_{p}$
has length $q+1$, that is, $z\mathcal{Z}_{p}=C\cup D$ where
$D\in\mathcal{D}(\mathfrak{S})$ and $|D|=q+1$. Thus, Theorem 3.10 shows that
$\mathfrak{S}$ is automorphic.

Using similar counting arguments, Corollary 3.6, and Theorem 3.10 again, we
see that $\mathfrak{S}$ is also automorphic if $\varphi(\mathfrak{S})$ is
symmetric. ∎

##  5 Connection between Schur rings over $\mathcal{Z}\times\mathcal{Z}_{n}$
and Difference Sets

Consider a primitive subset of $z\mathcal{Z}_{v}$ contained in a Schur ring
over $G=\mathcal{Z}\times\mathcal{Z}_{v}$ (we are assuming the discrete case,
as the symmetric case is similar), say $C=zD$. Say $|D|=k$. Suppose that the
Schur subring $T(\mathfrak{S})$ is trivial. Then
$\overline{C}\,\overline{C}^{*}\in T(\mathfrak{S})$, that is,
$\overline{C}\,\overline{C}^{*}=n+\lambda\overline{\mathcal{Z}_{v}}$ for some
integer $\lambda$ and $n=k-\lambda$. This implies that $D$ is a difference set
of $\mathcal{Z}_{v}$, as mentioned in the proof of Theorem 4.1. As the
primitive subset $C$ was arbitrary, it must hold that $\mathcal{Z}_{v}$ is a
union of disjoint difference sets in this case.

######  Definition 5.1.

We say that a partition $\mathcal{D}$ of a finite group $G$ is a _difference
partition_ if each block in $\mathcal{D}$ is itself a difference set.

There are many simple examples of difference partitions. For example,
$\\{\\{G\\}\\}$ and $\\{\\{g\\}\mid g\in G\\}$ are difference partitions, as
both $G$ and $\\{g\\}$ are trivial difference sets. Likewise,
$\\{\\{g\\},G\smallsetminus\\{g\\}\\}$ is a difference partition for any $g\in
G$. Generalizing this last example, let $D$ be any difference set of $G$, then
$\\{D,G\smallsetminus D\\}$ is a difference partition. Also,
$\\{D\\}\cup\\{\\{g\\}\mid g\in G\smallsetminus D\\}$ is another difference
partition. Finally, let $D$ be the set of quadratic residues in
$\mathcal{Z}_{p}$ for $p\equiv 3\pmod{4}$, the Paley difference set. Then
$\\{\\{1\\},D,D^{*}\\}$ is also a difference partition (this example can be
generalized using other Paley-Hadamard difference sets). Note that some of
these difference partitions are associated to Schur rings over
$\mathcal{Z}_{v}$.

Note that in all of the previous examples of difference partitions, except
maybe the complementary partition $\\{D,G\smallsetminus D\\}$, all of these
difference partitions involve a trivial difference set, usually a singleton.
Although the complementary partition $\\{D,G\smallsetminus D\\}$ does not
necessarily involve a trivial difference set over $G$, it is still quite
trivial as a partition.

######  Definition 5.2.

We say that a difference partition is _trivial_ if either it contains a
trivial difference set or contains exactly two blocks. Otherwise, we say that
a difference partition is _non-trivial_.

As hinted above, all the examples of difference partitions listed herein are
trivial. To provide an example of a non-trivial difference partition is more
challenging. First of all, translates of the same difference set always have
non-empty intersection, which makes translates unusable for forming a
partition. Automorphic images of difference sets are, of course, difference
sets but often are equal to translates of the original difference sets. This
theory of multipliers of a difference set is a well-studied topic. Thus, in
order for a group to have a difference partition, almost certainly it will
need at least two non-equivalent difference sets, a situation which is quite
rare (the existence of a non-trivial difference set of a group is itself a
fairly rare phenomenon). In the case of $\mathcal{Z}_{p}$, this is a
requirement as $\mathcal{Z}_{p}$ cannot be partitioned using blocks of all the
same size. The two smallest primes that even have two non-equivalent, non-
trivial difference sets are $p=31$ and $p=307$, neither of which have block
sizes that could form a non-trivial difference partition. It is natural to
even ask if there is a non-trivial difference partition.

######  Question 5.3.

Given a cyclic group of prime order $\mathcal{Z}_{p}$, does there exists a
non-trivial difference partition? How about over an arbitrary cyclic group? Or
an arbitrary abelian group?

The counting arguments used in the proof of Theorem 4.1 show that there is no
non-trivial difference partition over $\mathcal{Z}_{p}$ if $p$ is a Fermat or
a safe prime.

Of course, the partition of $z\mathcal{Z}_{v}$ in a Schur ring over
$\mathcal{Z}\times\mathcal{Z}_{v}$ requires more than a difference partition.
For example, if $zA,zB\in\mathcal{D}(\mathfrak{S})$, then
$(zA)(zB)^{*}=AB^{*}\subseteq\mathcal{Z}_{v}$. If $A\neq B$, then
$\overline{A}\,\overline{B}^{*}=\lambda\overline{\mathcal{Z}_{p}\smallsetminus
1}$. This implies that $|A||B|=\lambda(v-1)$, a similar formula to the classic
formula of difference sets, namely $k(k-1)=\lambda(v-1)$.

When $T(\mathfrak{S})$ is not trivial, similar properties of partitions on
$z\mathcal{Z}_{v}$ are required, and these partitions can be viewed as
generalizations of difference partitions. This is analogous to the fact that
Schur rings over $\mathcal{Z}_{p}$ in a way generalize difference sets over
$\mathcal{Z}_{p}$. For example, the Schur ring over $\mathcal{Z}_{p}$ which
corresponds to the unique automorphism of order $\dfrac{p-1}{2}$ consists of
three primitive sets. When $p\equiv 3\pmod{4}$, the non-indentity classes are
Paley difference sets. When $p\equiv 1\pmod{4}$, the non-identity classes are
reversible partial difference sets.

Due to the remarks and examples given above, the consideration of non-trivial
difference partitions will be necessary for further study of Schur rings over
$\mathcal{Z}\times\mathcal{Z}_{p}$. They will likely also be of broader
interest in the theory of Schur rings and algebraic combinatorics itself.

## References

  * [1] Nicholas Bastian, Jaden Brewer, Stephen Humphries, Andrew Misseldine, and Cache Thompson.  On Schur rings over infinite groups.  Algebras and Representation Theory, 2019.
  * [2] Wieb Bosma, John Cannon, and Catherine Playoust.  The Magma algebra system. I. The user language.  J. Symbolic Comput., 24(3-4):235–265, 1997.  Computational algebra and number theory (London, 1993).
  * [3] Stephen P. Humphries.  On schur rings over infinite groups ii.  preprint, pages 1–20, 2019.
  * [4] Ka Hin Leung and Shin Hing Man.  On Schur rings over cyclic groups II.  Journal of Algebra, 183:273–285, 1996.
  * [5] Ka Hin Leung and Shin Hing Man.  On Schur rings over cyclic groups.  Israel Journal of Mathematics, 106:251–267, 1998.
  * [6] W. R. Scott.  Group Theory.  Dover Publications, Inc., New York, 2nd edition edition, 1987.
  * [7] The OEIS Foundation Inc.  The on-line encyclopedia of integer sequences, 2021.
  * [8] The OEIS Foundation Inc.  The on-line encyclopedia of integer sequences, 2021.
  * [9] Helmut Wielandt.  Zur theorie der einfach transitiven permutationsgruppen II.  Math. Z., 52:384–393, 1949.
  * [10] Helmut Wielandt.  Finite Permutation Groups.  Academic Press, New York-London, 1964.