34}}$, G. Steyn${}^{\textrm{\scriptsize 66}}$, J.H.
Stiller${}^{\textrm{\scriptsize 95}}$, D. Stocco${}^{\textrm{\scriptsize
115}}$, P. Strmen${}^{\textrm{\scriptsize 38}}$, A.A.P.
Suaide${}^{\textrm{\scriptsize 122}}$, T. Sugitate${}^{\textrm{\scriptsize
47}}$, C. Suire${}^{\textrm{\scriptsize 52}}$, M.
Suleymanov${}^{\textrm{\scriptsize 16}}$, M. Suljic${}^{\textrm{\scriptsize
25}}$, R. Sultanov${}^{\textrm{\scriptsize 55}}$, M.
Šumbera${}^{\textrm{\scriptsize 85}}$, S. Sumowidagdo${}^{\textrm{\scriptsize
50}}$, A. Szabo${}^{\textrm{\scriptsize 38}}$, I.
Szarka${}^{\textrm{\scriptsize 38}}$, A.
Szczepankiewicz${}^{\textrm{\scriptsize 137}}$, M.
Szymanski${}^{\textrm{\scriptsize 137}}$, U. Tabassam${}^{\textrm{\scriptsize
16}}$, J. Takahashi${}^{\textrm{\scriptsize 123}}$, G.J.
Tambave${}^{\textrm{\scriptsize 22}}$, N. Tanaka${}^{\textrm{\scriptsize
131}}$, M. Tarhini${}^{\textrm{\scriptsize 52}}$, M.
Tariq${}^{\textrm{\scriptsize 18}}$, M.G. Tarzila${}^{\textrm{\scriptsize
79}}$, A. Tauro${}^{\textrm{\scriptsize 35}}$, G. Tejeda
Muñoz${}^{\textrm{\scriptsize 2}}$, A. Telesca${}^{\textrm{\scriptsize 35}}$,
K. Terasaki${}^{\textrm{\scriptsize 130}}$, C.
Terrevoli${}^{\textrm{\scriptsize 29}}$, B. Teyssier${}^{\textrm{\scriptsize
133}}$, J. Thäder${}^{\textrm{\scriptsize 75}}$, D.
Thakur${}^{\textrm{\scriptsize 49}}$, D. Thomas${}^{\textrm{\scriptsize
120}}$, R. Tieulent${}^{\textrm{\scriptsize 133}}$, A.
Tikhonov${}^{\textrm{\scriptsize 53}}$, A.R. Timmins${}^{\textrm{\scriptsize
125}}$, A. Toia${}^{\textrm{\scriptsize 61}}$, S.
Trogolo${}^{\textrm{\scriptsize 26}}$, G. Trombetta${}^{\textrm{\scriptsize
33}}$, V. Trubnikov${}^{\textrm{\scriptsize 3}}$, W.H.
Trzaska${}^{\textrm{\scriptsize 126}}$, T. Tsuji${}^{\textrm{\scriptsize
130}}$, A. Tumkin${}^{\textrm{\scriptsize 101}}$, R.
Turrisi${}^{\textrm{\scriptsize 109}}$, T.S. Tveter${}^{\textrm{\scriptsize
21}}$, K. Ullaland${}^{\textrm{\scriptsize 22}}$, A.
Uras${}^{\textrm{\scriptsize 133}}$, G.L. Usai${}^{\textrm{\scriptsize 24}}$,
A. Utrobicic${}^{\textrm{\scriptsize 132}}$, M. Vala${}^{\textrm{\scriptsize
56}}$, L. Valencia Palomo${}^{\textrm{\scriptsize 71}}$, S.
Vallero${}^{\textrm{\scriptsize 26}}$, J. Van Der
Maarel${}^{\textrm{\scriptsize 54}}$, J.W. Van Hoorne${}^{\textrm{\scriptsize
114}}$,${}^{\textrm{\scriptsize 35}}$, M. van Leeuwen${}^{\textrm{\scriptsize
54}}$, T. Vanat${}^{\textrm{\scriptsize 85}}$, P. Vande
Vyvre${}^{\textrm{\scriptsize 35}}$, D. Varga${}^{\textrm{\scriptsize 139}}$,
A. Vargas${}^{\textrm{\scriptsize 2}}$, M. Vargyas${}^{\textrm{\scriptsize
126}}$, R. Varma${}^{\textrm{\scriptsize 48}}$, M.
Vasileiou${}^{\textrm{\scriptsize 90}}$, A. Vasiliev${}^{\textrm{\scriptsize
81}}$, A. Vauthier${}^{\textrm{\scriptsize 72}}$, O. Vázquez
Doce${}^{\textrm{\scriptsize 96}}$,${}^{\textrm{\scriptsize 36}}$, V.
Vechernin${}^{\textrm{\scriptsize 135}}$, A.M. Veen${}^{\textrm{\scriptsize
54}}$, A. Velure${}^{\textrm{\scriptsize 22}}$, E.
Vercellin${}^{\textrm{\scriptsize 26}}$, S. Vergara
Limón${}^{\textrm{\scriptsize 2}}$, R. Vernet${}^{\textrm{\scriptsize 8}}$, M.
Verweij${}^{\textrm{\scriptsize 138}}$, L. Vickovic${}^{\textrm{\scriptsize
118}}$, J. Viinikainen${}^{\textrm{\scriptsize 126}}$, Z.
Vilakazi${}^{\textrm{\scriptsize 129}}$, O. Villalobos
Baillie${}^{\textrm{\scriptsize 103}}$, A. Villatoro
Tello${}^{\textrm{\scriptsize 2}}$, A. Vinogradov${}^{\textrm{\scriptsize
81}}$, L. Vinogradov${}^{\textrm{\scriptsize 135}}$, T.
Virgili${}^{\textrm{\scriptsize 30}}$, V. Vislavicius${}^{\textrm{\scriptsize
34}}$, Y.P. Viyogi${}^{\textrm{\scriptsize 136}}$, A.
Vodopyanov${}^{\textrm{\scriptsize 67}}$, M.A. Völkl${}^{\textrm{\scriptsize
95}}$, K. Voloshin${}^{\textrm{\scriptsize 55}}$, S.A.
Voloshin${}^{\textrm{\scriptsize 138}}$, G. Volpe${}^{\textrm{\scriptsize
33}}$,${}^{\textrm{\scriptsize 139}}$, B. von Haller${}^{\textrm{\scriptsize
35}}$, I. Vorobyev${}^{\textrm{\scriptsize 36}}$,${}^{\textrm{\scriptsize
96}}$, D. Vranic${}^{\textrm{\scriptsize 35}}$,${}^{\textrm{\scriptsize 99}}$,
J. Vrláková${}^{\textrm{\scriptsize 40}}$, B. Vulpescu${}^{\textrm{\scriptsize
71}}$, B. Wagner${}^{\textrm{\scriptsize 22}}$, J.
Wagner${}^{\textrm{\scriptsize 99}}$, H. Wang${}^{\textrm{\scriptsize 54}}$,
M. Wang${}^{\textrm{\scriptsize 7}}$, D. Watanabe${}^{\textrm{\scriptsize
131}}$, Y. Watanabe${}^{\textrm{\scriptsize 130}}$, M.
Weber${}^{\textrm{\scriptsize 35}}$,${}^{\textrm{\scriptsize 114}}$, S.G.
Weber${}^{\textrm{\scriptsize 99}}$, D.F. Weiser${}^{\textrm{\scriptsize
95}}$, J.P. Wessels${}^{\textrm{\scriptsize 62}}$, U.
Westerhoff${}^{\textrm{\scriptsize 62}}$, A.M.
Whitehead${}^{\textrm{\scriptsize 91}}$, J. Wiechula${}^{\textrm{\scriptsize
61}}$,${}^{\textrm{\scriptsize 94}}$, J. Wikne${}^{\textrm{\scriptsize 21}}$,
G. Wilk${}^{\textrm{\scriptsize 78}}$, J. Wilkinson${}^{\textrm{\scriptsize
95}}$, G.A. Willems${}^{\textrm{\scriptsize 62}}$, M.C.S.
Williams${}^{\textrm{\scriptsize 106}}$, B. Windelband${}^{\textrm{\scriptsize
95}}$, M. Winn${}^{\textrm{\scriptsize 95}}$, S.
Yalcin${}^{\textrm{\scriptsize 70}}$, P. Yang${}^{\textrm{\scriptsize 7}}$, S.
Yano${}^{\textrm{\scriptsize 47}}$, Z. Yin${}^{\textrm{\scriptsize 7}}$, H.
Yokoyama${}^{\textrm{\scriptsize 131}}$,${}^{\textrm{\scriptsize 72}}$, I.-K.
Yoo${}^{\textrm{\scriptsize 98}}$, J.H. Yoon${}^{\textrm{\scriptsize 51}}$, V.
Yurchenko${}^{\textrm{\scriptsize 3}}$, A. Zaborowska${}^{\textrm{\scriptsize
137}}$, V. Zaccolo${}^{\textrm{\scriptsize 82}}$, A.
Zaman${}^{\textrm{\scriptsize 16}}$, C. Zampolli${}^{\textrm{\scriptsize
106}}$,${}^{\textrm{\scriptsize 35}}$, H.J.C. Zanoli${}^{\textrm{\scriptsize
122}}$, S. Zaporozhets${}^{\textrm{\scriptsize 67}}$, N.
Zardoshti${}^{\textrm{\scriptsize 103}}$, A.
Zarochentsev${}^{\textrm{\scriptsize 135}}$, P. Závada${}^{\textrm{\scriptsize
57}}$, N. Zaviyalov${}^{\textrm{\scriptsize 101}}$, H.
Zbroszczyk${}^{\textrm{\scriptsize 137}}$, I.S. Zgura${}^{\textrm{\scriptsize
59}}$, M. Zhalov${}^{\textrm{\scriptsize 87}}$, H.
Zhang${}^{\textrm{\scriptsize 22}}$,${}^{\textrm{\scriptsize 7}}$, X.
Zhang${}^{\textrm{\scriptsize 7}}$,${}^{\textrm{\scriptsize 75}}$, Y.
Zhang${}^{\textrm{\scriptsize 7}}$, C. Zhang${}^{\textrm{\scriptsize 54}}$, Z.
Zhang${}^{\textrm{\scriptsize 7}}$, C. Zhao${}^{\textrm{\scriptsize 21}}$, N.
Zhigareva${}^{\textrm{\scriptsize 55}}$, D. Zhou${}^{\textrm{\scriptsize 7}}$,
Y. Zhou${}^{\textrm{\scriptsize 82}}$, Z. Zhou${}^{\textrm{\scriptsize 22}}$,
H. Zhu${}^{\textrm{\scriptsize 7}}$,${}^{\textrm{\scriptsize 22}}$, J.
Zhu${}^{\textrm{\scriptsize 115}}$,${}^{\textrm{\scriptsize 7}}$, A.
Zichichi${}^{\textrm{\scriptsize 12}}$,${}^{\textrm{\scriptsize 27}}$, A.
Zimmermann${}^{\textrm{\scriptsize 95}}$, M.B.
Zimmermann${}^{\textrm{\scriptsize 62}}$,${}^{\textrm{\scriptsize 35}}$, G.
Zinovjev${}^{\textrm{\scriptsize 3}}$, M. Zyzak${}^{\textrm{\scriptsize 42}}$

## Affiliation notes

0Deceased

idp1824064Also at: Georgia State University, Atlanta, Georgia, United States

idp3239488Also at: Also at Department of Applied Physics, Aligarh Muslim
University, Aligarh, India

idp3964800Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn
Institute of Nuclear, Physics, Moscow, Russia

## Collaboration Institutes

1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
4Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
5Budker Institute for Nuclear Physics, Novosibirsk, Russia
6California Polytechnic State University, San Luis Obispo, California, United
States
7Central China Normal University, Wuhan, China
8Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France
9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and
Mérida, Mexico
12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico
Fermi’, Rome, Italy
13Chicago State University, Chicago, Illinois, United States
14China Institute of Atomic Energy, Beijing, China
15Commissariat à l’Energie Atomique, IRFU, Saclay, France
16COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan
17Departamento de Física de Partículas and IGFAE, Universidad de Santiago de
Compostela, Santiago de Compostela, Spain
18Department of Physics, Aligarh Muslim University, Aligarh, India
19Department of Physics, Ohio State University, Columbus, Ohio, United States
20Department of Physics, Sejong University, Seoul, South Korea
21Department of Physics, University of Oslo, Oslo, Norway
22Department of Physics and Technology, University of Bergen, Bergen, Norway
23Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN, Rome,
Italy
24Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
25Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
26Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
30Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy
33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy
34Division of Experimental High Energy Physics, University of Lund, Lund,
Sweden
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Excellence Cluster Universe, Technische Universität München, Munich, Germany
37Faculty of Engineering, Bergen University College, Bergen, Norway
38Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia
39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
40Faculty of Science, P.J. Šafárik University, Košice, Slovakia
41Faculty of Technology, Buskerud and Vestfold University College, Tonsberg,
Norway
42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität
Frankfurt, Frankfurt, Germany
43Gangneung-Wonju National University, Gangneung, South Korea
44Gauhati University, Department of Physics, Guwahati, India
45Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
46Helsinki Institute of Physics (HIP), Helsinki, Finland
47Hiroshima University, Hiroshima, Japan
48Indian Institute of Technology Bombay (IIT), Mumbai, India
49Indian Institute of Technology Indore, Indore, India
50Indonesian Institute of Sciences, Jakarta, Indonesia
51Inha University, Incheon, South Korea
52Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud, CNRS-
IN2P3, Orsay, France
53Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
54Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands
55Institute for Theoretical and Experimental Physics, Moscow, Russia
56Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia
57Institute of Physics, Academy of Sciences of the Czech Republic, Prague,
Czech Republic
58Institute of Physics, Bhubaneswar, India
59Institute of Space Science (ISS), Bucharest, Romania
60Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
61Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
62Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster,
Germany
63Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
64Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico
65Institut Pluridisciplinaire Hubert Curien (IPHC), Université de Strasbourg,
CNRS-IN2P3, Strasbourg, France
66iThemba LABS, National Research Foundation, Somerset West, South Africa
67Joint Institute for Nuclear Research (JINR), Dubna, Russia
68Konkuk University, Seoul, South Korea
69Korea Institute of Science and Technology Information, Daejeon, South Korea
70KTO Karatay University, Konya, Turkey
71Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université
Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France
72Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS-IN2P3, Grenoble, France
73Laboratori Nazionali di Frascati, INFN, Frascati, Italy
74Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy
75Lawrence Berkeley National Laboratory, Berkeley, California, United States
76Moscow Engineering Physics Institute, Moscow, Russia
77Nagasaki Institute of Applied Science, Nagasaki, Japan
78National Centre for Nuclear Studies, Warsaw, Poland
79National Institute for Physics and Nuclear Engineering, Bucharest, Romania
80National Institute of Science Education and Research, Bhubaneswar, India
81National Research Centre Kurchatov Institute, Moscow, Russia
82Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
83Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam, Netherlands
84Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom
85Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež u
Prahy, Czech Republic
86Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
87Petersburg Nuclear Physics Institute, Gatchina, Russia
88Physics Department, Creighton University, Omaha, Nebraska, United States
89Physics Department, Panjab University, Chandigarh, India
90Physics Department, University of Athens, Athens, Greece
91Physics Department, University of Cape Town, Cape Town, South Africa
92Physics Department, University of Jammu, Jammu, India
93Physics Department, University of Rajasthan, Jaipur, India
94Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen,
Germany
95Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
96Physik Department, Technische Universität München, Munich, Germany
97Purdue University, West Lafayette, Indiana, United States
98Pusan National University, Pusan, South Korea
99Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung, Darmstadt, Germany
100Rudjer Bošković Institute, Zagreb, Croatia
101Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
102Saha Institute of Nuclear Physics, Kolkata, India
103School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
104Sección Física, Departamento de Ciencias, Pontificia Universidad Católica
del Perú, Lima, Peru
105Sezione INFN, Bari, Italy
106Sezione INFN, Bologna, Italy
107Sezione INFN, Cagliari, Italy
108Sezione INFN, Catania, Italy
109Sezione INFN, Padova, Italy
110Sezione INFN, Rome, Italy
111Sezione INFN, Trieste, Italy
112Sezione INFN, Turin, Italy
113SSC IHEP of NRC Kurchatov institute, Protvino, Russia
114Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
115SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3,
Nantes, France
116Suranaree University of Technology, Nakhon Ratchasima, Thailand
117Technical University of Košice, Košice, Slovakia
118Technical University of Split FESB, Split, Croatia
119The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Cracow, Poland
120The University of Texas at Austin, Physics Department, Austin, Texas,
United States
121Universidad Autónoma de Sinaloa, Culiacán, Mexico
122Universidade de São Paulo (USP), São Paulo, Brazil
123Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
124Universidade Federal do ABC, Santo Andre, Brazil
125University of Houston, Houston, Texas, United States
126University of Jyväskylä, Jyväskylä, Finland
127University of Liverpool, Liverpool, United Kingdom
128University of Tennessee, Knoxville, Tennessee, United States
129University of the Witwatersrand, Johannesburg, South Africa
130University of Tokyo, Tokyo, Japan
131University of Tsukuba, Tsukuba, Japan
132University of Zagreb, Zagreb, Croatia
133Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne,
Lyon, France
134Università di Brescia, Brescia, Italy
135V. Fock Institute for Physics, St. Petersburg State University, St.
Petersburg, Russia
136Variable Energy Cyclotron Centre, Kolkata, India
137Warsaw University of Technology, Warsaw, Poland
138Wayne State University, Detroit, Michigan, United States
139Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary
140Yale University, New Haven, Connecticut, United States
141Yonsei University, Seoul, South Korea
142Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule
Worms, Worms, Germany


# Sublinear biLipschitz equivalence and sublinearly Morse boundaries

Gabriel Pallier  Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
<EMAIL_ADDRESS>and  Yulan Qing  Shanghai Center for Mathematical
Sciences, Fudan University, Shanghai, China<EMAIL_ADDRESS>
###### Abstract.

A sublinear biLipschitz equivalence (SBE) between metric spaces is a map from
one space to another that distorts distances with bounded multiplicative
constants and sublinear additive error. Given any sublinear function $\kappa$,
$\kappa$-Morse boundaries are defined for all geodesic proper metric spaces as
a quasi-isometrically invariant and metrizable topological space of quasi-
geodesic rays. In this paper, we prove that $\kappa$-Morse boundaries of
proper geodesic metric spaces are invariant under suitable SBEs. A tool in the
proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings
of the half line, generalizing quasi-geodesic rays. As an application we
distinguish a pair of right-angled Coxeter groups brought up by Behrstock up
to sublinear biLipschitz equivalence. We also show that under mild
assumptions, generic random walks on countable groups are sublinear rays.

G.P. gratefully acknowledges funding by the DFG 281869850 (RTG 2229) and the
ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’.

##  1\. Introduction

Sublinear biLipschitz equivalence is an equivalence relation between metric
spaces that naturally generalizes quasiisometries. It appeared first without a
name, and then more explicitly in the work of Cornulier on asymptotic cones of
Lie groups [Cor08, Cor11], before being studied for its own sake [Cor19,
Pal20]. Following Cornulier we will abbreviate this relation as SBE. Because
Gromov hyperbolicity admits a characterization in terms of asymptotic cones,
it follows that being hyperbolic is an SBE-invariant property among compactly
generated locally compact groups; this was noted by Cornulier [Cor19, Theorem
4.3].

In this project we extend this result beyond groups and spaces that are Gromov
hyperbolic. Precisely, given a proper geodesic space that is not Gromov
hyperbolic, but exhibiting some features of Gromov hyperbolic spaces, one can
study its large scale hyperbolic-like structure by describing the _sublinearly
Morse boundaries_ of the group [QRT20]. The latter is a topological space
collecting a large set of quasi-geodesic rays behaving in a hyperbolic
fashion; when the space is Gromov-hyperbolic, all of its sublinearly Morse
boundaries are homeomorphic to its Gromov boundary. In this paper we show that
the set of subblinearly Morse directions are invariant under suitable
sublinear biLipschitz equivalence. We then go further to obtain simultaneous
generalizations of two distinct previous results in the literature :

  1. (1)

(See §1.1 below) Qing, Rafi and Tiozzo's theorem that the homeomorphism type
of the sublinearly Morse boundary is a quasi-isometry invariant among proper
geodesic metric spaces [QRT20, Theorem A(2)].

  2. (2)

(See §1.2 below) Cornulier's theorem that the homeomorphism type of the Gromov
boundary is SBE invariant among Gromov-hyperbolic groups [Cor19].

We recall some context on these two theorems.

###  1.1. Boundaries

In his seminal article [Gro87], Gromov introduced the class of hyperbolic
groups and attached to such groups an equivariant bordification, now called
the Gromov boundary. The class of Gromov-hyperbolic group is closed under
quasiisometry, and the quasiisometries extend equivariantly to the Gromov
boundaries.

The class of Gromov hyperbolic groups and spaces is however not vast enough to
include natural examples such as $\mathrm{CAT}(0)$ groups and mapping class
groups of surfaces of finite type.

For $\mathrm{CAT}(0)$ groups and spaces, the visual boundary (the set of all
geodesic rays emanating from a fixed base-point $(X,{\mathfrak{o}})$, up to
fellow travel) does not provide a good large-scale invariant (as indicated by
the works from Croke-Kleiner [CK00] to Qing [Qin16]). In [QRT19], the second
named author and Rafi consider the set of quasi-geodesic rays whose Morse
property is weakened compared to that of geodesic rays in Gromov spaces. In
particular, given a sublinear function $\kappa$, Qing-Rafi define a quasi-
geodesic ray $\gamma$ to be _sublinearly $\kappa$-Morse_ if any other geodesic
segment with endpoints on $\gamma$ is uniformly $\kappa$-close to $\gamma$,
i.e. their distances to $\gamma$ is bounded above by $n({\sf q},{\sf
Q})\kappa(\lVert x\rVert)$, where $n({\sf q},{\sf Q})$ is a constant depends
only on the quasi-geodesic constants $({\sf q},{\sf Q})$ of the segment, and
the distance of each point on the segment to the origin, $\lVert
x\rVert=d({\mathfrak{o}},x)$. The collection of all such quasi-geodesic rays,
together with a coarse cone topology, is referred to as the $\kappa$-boundary
of $X$, and denoted ${\partial_{\kappa}}X$. These boundaries are shown to be
quasi-isometrically invariant topological spaces attached to all proper
geodesic spaces [QRT20]. Therefore one can denote a $\kappa$-boundary of a
group with ${\partial_{\kappa}}G$. Furthermore, they are metrizable
topological spaces ([QRT20]). Since their introduction, sublinearly Morse
boundaries are studied and compared to Gromov boundaries in various ways, such
as via visibility, divergence and contracting properties (See[MQZ21], [IZ] and
[Zal21] ).

One important application of the sublinear boundaries is that, for
appropriately chosen $\kappa$, ${\partial_{\kappa}}G$ is a topological model
for the Poisson boundaries of simple random walks on various groups, such as
right-angled Artin groups [QRT19], mapping class groups and relative
hyperbolic groups [QRT20], hierarchically hyperbolic groups [NQ22], CAT(0)
groups [GQR22] and Teichmüller spaces [GQR22]. The sublinearly Morse
directions are also shown to be generic in Patterson Sullivan measure under
suitable conditions [GQR22]. Most recently, Choi [Cho] claim this result to
hold for all groups with two independent isometries with contracting axes.

###  1.2. Sublinear biLipschitz equivalence

Sublinear biLipschitz equivalence (SBE) appeared in works of Cornulier, where
it was motivated by the quasiisometry classification of connected Lie groups.
Cornulier noted that while the quasiisometry classification of all such groups
reduces to that of closed subgroups of real upper triangular matrices, the
sublinear biLipschitz equivalence classification reduces to that of a smaller
class, and that it was completely treated by the literature in the nilpotent
case [Cor11]. In [Cor19] Cornulier asked about the SBE classification for
other classes of groups, especially the word-hyperbolic groups.

As mentioned above, the first SBE invariant is the asymptotic cone; it is also
the most natural one, since SBE may be defined as the largest class of maps
inducing biLipschitz homeomorphisms between asymptotic cones with fixed base-
points [Cor11]. A geodesic metric space $X$ is Gromov-hyperbolic if and only
if all its asymptotic cones are real trees, for every choice of sequence of
base-points [Gro93, 2.A] [Dru02, Proposition 3.A.4]. It follows that Gromov-
hyperbolicity is an SBE-invariant among compactly generated locally compact
groups, and especially among finitely generated groups)[Cor19, Theorem 4.3].
Cornulier proved that SBEs between Gromov-hyperbolic groups induce biHölder
homeomorphisms between their Gromov boundaries; this was slightly improved by
the first named author showing that a sublinear conformal structure in the
Gromov boundary is actually preserved [Cor19, Pal20]. In this paper we extend
the topological invairance of the Gromov boundary to that of a family of
topological spaces that can be attached to any proper geodesic metric spaces:

######  Theorem A.

(Theorem 4.4) Let $L\geqslant 1$, and let $\theta$ be a concave, nondecreasing
and stictly sublinear function. Consider proper geodesic metric spaces $X$ and
$Y$, let $\Phi\colon\thinspace X\to Y$ be a $(L,\theta)$-sublinear bi-
Lipschitz equivalence between $X$ and $Y$. Then, for every concave,
nondecreasing and strictly sublinear function $\kappa$ that dominates
$\theta$, there is a homeomorphism
$\Phi_{\star}\colon\thinspace\partial_{\kappa}X\to\partial_{\kappa}Y$, induced
by $\Phi$.

In the statement when we say that $\kappa$ dominates $\theta$ we mean that
there exists constants $C_{1},C_{2}$ and a nonnegative real number $t_{0}$
such that for all $t>t_{0}$, $\kappa(t)\geq C_{1}\theta(t)+C_{2}$. For the
complete definitions of the $\kappa$-Morse boundary $\partial_{\kappa}$ and of
a $(L,\theta)$-sublinear biLipschitz equivalence we refer to Subsection 2.4
and Definition 2.11 respectively. When $\theta=1$, $\Phi$ is a quasiisometry
and Theorem A is exactly [QRT20, Theorem A(2)]; when $X$ and $Y$ are Gromov
hyperbolic, Theorem A is a consequence of Theorem 1.7 in [Cor19].

### Application

As an application, we use Theorem A to distinguish two non-relatively
hyperbolic right-angled Coxeter groups up to SBE in Section 4. This pair was
presented by Behrstock in [Be19]. Determining whether these two groups have
bilipschitz homeomorphic asymptotic cones or not is an open question [Be]. On
the other hand, we check in 4.2 that their sublinearly Morse boundaries have
different topological dimensions, which allows us to conclude from Theorem A
that they cannot be sublinear biLipschitz equivalent.

The proof of Theorem 4.4 makes use of sublinear rays, introduced in [Pal20]
(there called $(\lambda,O(v))$-rays). These are images of the half-line under
a sublinear bi-Lipschitz embedding. As such, they are analogues of quasi-
geodesics, but with an additive constant of quasigedoesicity that grows
sublinearly. If the growth if logarithmic, we call the resulting sublinear ray
a logarithmic ray. Sublinear rays also play an important role in the statement
of our second result.

###  1.3. Random walks

Similar to the development of sublinearly Morse boundaries, an important
motivation behind this project stems from simple random walk on finitely
generated groups. In [Tio15], Tiozzo show that given a surface $S$, in the
Teichmüller space $\mathcal{T}(S)$, a generic random walk tracks a geodesic
sublinearly. In Section 5 of this paper, we show that this implies that a
generic random walk is a sublinear ray.

######  Theorem B (Theorem 5.1).

Let $G$ be the mapping class group $\operatorname{Mod}(S)$ of a finite type
surface, or let $G$ be a finitely generated relatively hyperbolic group. Let
$\mu$ be a probability measure on $G$ with finite first moment with respect to
any word metric on $G$, such that the subsemigroup generated by the support of
$\mu$ is a non-amenable group. Then almost every sample path of the random
walk associated to $\mu$ is a logarithmic ray.

In fact, a version of this result with a weaker conclusion holds for a larger
class of group actions: we combine the proof of Theorem 5.1 together Theorem 6
in [Tio15] to obtain the following.

######  Theorem B' (Theorem 5.2).

Let $G$ be a countable group acting via isometries on a proper, geodesic,
metric space $(X,d)$ with a non-trivial, stably visible compactification. Let
${\mathfrak{o}}$ denote a basepoint in $X$. Let $\mu$ be a probability measure
on G with finite first moment with respect to $d$, such that the subsemigroup
generated by the support of $\mu$ is a non-amenable group. Then for almost
every sample path $(w_{n})$ of the random walk associated to $\mu$,
$(w_{n}\cdot{\mathfrak{o}})$ is a sublinear ray.

While Theorem 5.1 is logically independent from Theorem 4.4, both theorems
promote the use of sublinear rays. Whereas a generic random walk is not a
quasi-geodesic, it is a sublinear ray, and many of the geometric techniques
devised for quasi-geodesics can be employed to treat sublinear rays as if they
were quasi-geodesics.

###  1.4. Organization of the paper

Section 2 collects preliminary information; especially we recall the relevant
definitions and facts concerning sublinear biLipschitz equivalence, rays, and
the sublinearly Morse boundaries. Beware that we chose to adopt the notation
from [QRT20] so that the notation for sublinear biLipschitz equivalence is not
the usual one. Section 3 is a preparation for Section 4, which is itself
devoted to the proof of Theorem 4.4. Section 5 deals with Theorem B and
Theorem B’. It only builds on the preliminaries and can be read without
Sections 3 and 4.

##  2\. Preliminaries

###  2.1. Notation and convention for the sublinear functions

Throughout this paper, let $X$ and $Y$ denote pointed proper geodesic metric
spaces. The base-points in both spaces are denoted $\mathfrak{o}$. The
distance to the base-point is denoted $\ \|x\|=d(x,\mathfrak{o})$ for all
$x\in X$ or $x\in Y$. Let $\kappa$ be a concave nondecreasing and strictly
sublinear function. The last condition means that $\kappa(r)/r$ goes to $0$ as
$r$ tends to $+\infty$. We also assume $\kappa\geqslant 1$.

By $\kappa(x)$ for $x\in X$ or $Y$ we mean $\kappa(\|x\|)$. Let $Z\subseteq X$
be a closed subspace of $X$ and ${\sf D}>0$, then $\mathcal{N}_{\kappa}(Z,{\sf
D})$ will denote $\\{x\in X:d(x,Z)\leqslant{\sf D}\kappa(x)\\}$. We say that
${\sf D}$ is small with respect to $r$, and write ${\sf D}\ll r$, if ${\sf
D}\leqslant r/(2\kappa(r))$.

###  2.2. Sublinear estimates

Here is a basic sublinear estimate that we need:

######  Lemma 2.1 (Sublinear Estimation Lemma).

For any ${\sf D}_{0}>0$, there exists ${\sf D}_{1},{\sf D}_{2}>0$ depending on
${\sf D}_{0}$ and $\kappa$ so that, for $x,y\in X$,

$d(x,y)\leq{\sf D}_{0}\cdot\kappa(x)\Longrightarrow{\sf
D}_{1}\kappa(x)\leq\kappa(y)\leq{\sf D}_{2}\kappa(x).$

###### Proof.

Since $\kappa$ is sublinear, there is $R>0$ such that
$\kappa(x)\leqslant\frac{1}{2{\sf D}_{0}}\|x\|$ as soon as $\|x\|\geqslant R$.
And then $d(x,y)\leqslant{\sf D}_{0}\kappa(x)$ implies that $\|y\|\leqslant
3\|x\|/2$ by the triangle inequality, so that, in all cases,

$\kappa(y)\leqslant\left[\sup_{r\geqslant
R}\frac{\kappa(3r/2)}{\kappa(r)}+\kappa(3R/2)\right]\kappa(x)$

where we used that $\kappa(x)\geqslant 1$. We may define ${\sf
D}_{2}=\sup_{r\geqslant R}\frac{\kappa(3r/2)}{\kappa(r)}+\kappa(3R/2)$. On the
other hand, if $\|x\|\geqslant R$ then $\|y\|\geqslant\|x\|/2$ so that

$\kappa(y)\geqslant\left[\inf_{r\geqslant
R}\frac{\kappa(r/2)}{\kappa(r)}\right]\kappa(x)$

Setting ${\sf D}_{1}=\min(\inf_{r\geqslant
R}\frac{\kappa(r/2)}{\kappa(r)},1/\kappa(R))$ finishes the proof. ∎

###  2.3. Quasi-geodesics and $\theta$-rays

Here, $\theta$ is a function with the same properties as $\kappa$, that were
specified in §2.1.

######  Definition 2.2 (Sublinear ray).

Let $X$ be a proper pointed geodesic metric space. Let $L\geqslant 1$ be a
constant. Say that $\gamma:[0,+\infty)\to X$ with $\gamma(0)=\mathfrak{o}$ is
a $(L,\theta)$-ray, or a $\theta$-ray for short, if for every
$s,t\in[0,+\infty)$

(2.1) $\frac{1}{L}|s-t|-\theta(\max(s,t))\leqslant
d(\gamma(s),\gamma(t))\leqslant L|s-t|+\theta(\max(s,t)).$

If we do not want to specify $\theta$, we will just say that $\gamma$ as in
the definition above is a sublinear ray. Beware that in [Pal20] we did not ask
$\gamma(0)=\mathfrak{o}$ in the definition of a sublinear ray but we do it
here. The difference is not a serious one, as one may simply advance the
function $\theta$ (i.e. replace $\theta$ with $\theta(\mathsf{D}+\cdot)$) to
accommodate for the change.

######  Remark 2.3 (compare [Pal20, Lemma 3.2]).

If $\gamma$ is a sublinear ray with large-scale Lipschitz constant $L$ and
sublinear function $\theta$, then for $s$ large enough
$\frac{1}{2L}|s|\leqslant\|\gamma(s)\|\leqslant 2L|s|$ (apply (2.1) with fixed
$t$), hence there exists $\widehat{\theta}=O(\theta)$ such that for $s$ and
$t$ large enough,

(2.2)
$\frac{1}{L}|s-t|-\widehat{\theta}(\max(\|\gamma(s)\|,\|\gamma(t)\|))\leqslant
d(\gamma(s),\gamma(t))\leqslant
L|s-t|+\widehat{\theta}(\max(\|\gamma(s)\|,\|\gamma(t)\|)).$

When $\theta$ is a constant, the Definition 2.2 is that of a quasi-geodesic
ray. For our purposes, it is however necessary to treat the latter
specifically because they play a special role in the definition of the
sublinearly Morse boundary. Hence, we will use $q$ for $L$ and $Q$ for
$\theta$ when we want to denote a quasi-geodesic ray.

######  Lemma 2.4 (Connect-the-dots for $\theta$-rays).

Let $\gamma$ be a $(L,\theta)$-ray in a proper geodesic metric space $X$. Then
there exists $n>0$ and $\widehat{\gamma}$ which is a $(L,n\cdot\theta)$-ray in
$X$ with the property that

  * •

$\gamma(t)=\widehat{\gamma}(t)$ for all nonnegative integer $t$.

  * •

$\widehat{\gamma}$ is continuous.

Moreover, there exists $n>0$ such that

(2.3) $d(\gamma(t),\widehat{\gamma}(t))\leqslant n\cdot\theta(t)$

for all $t$.

We will refer to $\widehat{\gamma}$ as a continuous completion of $\gamma$.

###### Proof.

For every $t\in\mathbf{N}$, choose a geodesic segment $\sigma_{t}$ from
$\gamma(t)$ to $\gamma(t+1)$ at unit speed and denote its length $\ell_{t}$.
Note that $\ell(t)\leqslant L+\theta(t+1)$ by the inequality on the right in
(2.1). Now for all $t\in[0,+\infty)$ set
$\widehat{\gamma}(t)=\sigma_{t}(\ell_{t}\cdot\\{t\\})$ where $\\{t\\}$ denotes
the fractional part of $t$. In this way $\widehat{\gamma}$ is continuous by
construction. Let $t,s\in[0,+\infty)$ be such that $t\leqslant s$. Then,
either $\lfloor s\rfloor=\lfloor t\rfloor$, in which case
$d(\widehat{(}\gamma(t),\gamma(s))\leqslant\ell_{\lfloor
t\rfloor}\leqslant\theta(t+1)\leqslant n_{0}\theta(t)$ for some $n_{0}$ by the
properties of $\theta$, or

$\displaystyle d(\gamma(t),\gamma(s))$ $\displaystyle\leqslant
d(\gamma(t),\gamma(\lceil t\rceil))+d(\gamma(\lceil t\rceil),\gamma(\lfloor
s\rfloor))+d(\gamma(\lfloor s\rfloor),\gamma(s))$ $\displaystyle\leqslant
L(\lfloor s\rfloor-\lceil t\rceil))+2\theta(s+1)$ $\displaystyle\leqslant
L|s-t|+2\theta(s+1).$

and

$\displaystyle d(\gamma(t),\gamma(s))$ $\displaystyle\geqslant d(\gamma(\lceil
s\rceil),\gamma(\lfloor t\rfloor))-d(\gamma(s),\gamma(\lceil
s\rceil))-d(\gamma(\lfloor t\rfloor),\gamma(t))$ $\displaystyle\geqslant
L^{-1}(\lceil s\rceil-\lfloor t\rfloor))-2\theta(s+1)$ $\displaystyle\geqslant
L^{-1}|s-t|+2\theta(s+1).$

Finally, $2\theta(s+1)\leqslant n_{1}\theta(s)$ for some $n_{1}$, it remains
to set $n=\max(n_{0},n_{1})$. ∎

######  Definition 2.5.

Let $\alpha,\beta$ be quasi-geodesic rays or $(L,\theta)$-rays for some $L$
and $\theta$ in a proper geodesic space. We say $\alpha\sim\beta$ if either of
the following holds:

  1. (1)

$\lim_{t\to\infty}\frac{d(\alpha(t),\beta)}{t}=0.$

  2. (2)

$\lim_{t\to\infty}\frac{d(\beta(t),\alpha)}{t}=0.$

######  Lemma 2.6.

(1) and (2) are equivalent.

Beware that Lemma 2.6 is false if one replaces $\alpha$ and $\beta$ with
arbitrary maps, even proper maps from the half-line to $X$ that respect the
inequality on the right hand side of (2.1). For instance, if one considers the
plane parametric curve $\beta$ parametrized by arc-length progressing along
the horizontal axis and making jumps of height $2^{n}$ at time $2^{n}$ for all
$n\geqslant 0$, and the parametrization of the horizontal axis $\alpha$, then
this pair has (1) but not (2).

###### Proof.

Assume (1) holds and define $\eta(s)=d(\alpha(s),\beta)$ for
$s\in[0,+\infty)$; this function $\eta$ is sublinear. Let us introduce the set

$\mathcal{T}=\left\\{t\in[0,+\infty):\exists
s\in[0,+\infty),d(\alpha(s),\beta(t))\leqslant 2d(\alpha(s),\beta)\right\\}.$

We claim that $\mathcal{T}$ cannot have large holes, more precisely there is
no $\varepsilon>0$ and sequence $(t_{n})$ with limit $+\infty$ such that

(2.4) $(t_{n}(1-\varepsilon);t_{n}(1+\varepsilon))\cap\mathcal{T}=\emptyset$

for all $n$. Indeed, assume the contrary and define

$s_{n}=\sup\\{s:\exists t<t_{n},d(\alpha(s),\beta(t))\leqslant
2d(\alpha(s),\beta)\\}$

It can be checked that $s_{n}$ is well defined for all $n$ and tends to
$+\infty$, since $\alpha$ is proper while $\beta[0,t_{n}]$ is bounded and $X$
is proper. Now consider a nearest point projection $p_{n}$ of
$\alpha(s_{n}+1)$ on $\beta$. Necessarily, $t>t_{n}(1+\varepsilon)$ for every
$t$ such that $p_{n}=\beta(t)$, in view of the definition of $s_{n}$ and
$t_{n}$. Let $q_{n}$ be a nearest-point projection of $\alpha(s_{n})$ on
$\beta$; by the same argument, if $q_{n}=\beta(t)$ then
$t<t_{n}(1-\varepsilon)$. Now, on the one hand by the triangle inequality,

$\displaystyle d(p_{n},q_{n})$ $\displaystyle\leqslant
d(p_{n},\alpha(s_{n}+1)+d(\alpha(s_{n}+1),\alpha(s_{n}))+d(\alpha(s_{n}),q_{n}))$
(2.5) $\displaystyle\leqslant L+\theta(s_{n+1})+\eta(s_{n})+\eta(s_{n}+1).$

On the other hand, by the left-hand side of (2.1),

(2.6) $\displaystyle d(p_{n},q_{n})$ $\displaystyle\geqslant
2L^{-1}\varepsilon t_{n}-\widehat{\theta}(\max(\|p_{n}\|,\|q_{n}\|))$

where $\widehat{\theta}=O(\theta)$. However, there is a constant $M>0$ such
that $M^{-1}t_{n}\leqslant s_{n}\leqslant Mt_{n}$ for $n$ large enough; one
can take $M=2L^{2}(1+\varepsilon)$, by considering the first inequality in
Remark 2.3.

Making $n\to+\infty$ and using that $\theta$, $\widehat{\theta}$ and $\eta$
are sublinear, (2.5) and (2.6) are in contradiction with one another. Thus
(2.4) cannot be true. It follows that there is a sublinear function $\mu$ such
that for all $t\geqslant 0$, there is $t^{\prime}$ with
$|t-t^{\prime}|\leqslant\mu(t)$ and $t^{\prime}\in\mathcal{T}$. And then
$d(\beta(t),\alpha)\leqslant
d(\beta(t),\beta(t^{\prime}))+d(\beta(t^{\prime}),\alpha)$, which is bounded
above by a sublinear function of $t$ involving $\eta$, $\mu$ and $L$ in view
of the definition of $\mathcal{T}$ and the linear control between $s$ and $t$
when $\beta(t^{\prime})$ is a closest point projection of $\alpha(s)$ on
$\beta$. We proved that (1) implies (2) for $(L,\theta)$ rays; the converse
implication holds by symmetry, and quasi-geodesics are $\theta$-rays, hence
the Lemma is proved. ∎

$\alpha$$\beta$$\beta(t_{n-1})$$\beta(t_{n})$$\alpha(s_{n})$$\alpha(s_{n}+1)$$p_{n}$$q_{n}$$\beta(t_{n}(1+\varepsilon))$$\beta(t_{n}(1-\varepsilon))$
Figure 1. Proof of Lemma 2.6.

###  2.4. Definition of the sublinearly Morse boundary $\partial X$

For more extensive references on sublinearly Morse boundaries, see [QRT19] and
[QRT20]. A $(\mathsf{q},\mathsf{Q})$-quasigeodesic is a map
$\gamma:[0,+\infty)\to X$ such that
$\frac{1}{\mathsf{q}}d(x,y)-\mathsf{Q}\leqslant
d(\gamma(x),\gamma(y))\leqslant\mathsf{q}d(x,y)+\mathsf{Q}$ for all
$x,y\in[0,+\infty)$.

######  Definition 2.7 ($\kappa$-Morse quasigeodesic [QRT20, Definition 3.2]).

Let $Z\subseteq X$ be a closed subspace. Let
$m_{Z}:\mathbb{R}^{2}\to\mathbb{R}$ be a proper function. We say that $Z$ is
$\kappa$-Morse with Morse gauge $m_{Z}$ if for every sublinear function
$\kappa^{\prime}$, for every $r>0,n>0$ such that
$m_{Z}(\mathsf{q},\mathsf{Q})\ll_{\kappa}r$, there exists
$R(Z,r,n,\kappa^{\prime})>0$ such that if $\beta$ is a
$(\mathsf{q},\mathsf{Q})$-quasigeodesic then

$d_{X}(\beta_{R},Z)\leqslant
n\cdot\kappa^{\prime}(R)\implies\beta|_{r}\subset\mathcal{N}_{\kappa}(Z,m_{Z}(\mathsf{q},\mathsf{Q})).$

Applying the definition to all $r>0$ we have the following alternative
characterization:

######  Definition 2.8.

Let $Z$ be a closed subspace. Let $m_{Z}:\mathbb{R}^{2}\to\mathbb{R}$ be a
proper function. We say that $Z$ is $\kappa$-Morse with Morse gauge $m_{Z}$ if
$\beta$ is any other $(\mathsf{q},\mathsf{Q})$-quasi-geodesic ray that
sublinearly tracks $Z$, then we have:

$\beta\in\mathcal{N}_{\kappa}(Z,m_{Z}(\mathsf{q},\mathsf{Q})).$

######  Proposition 2.9 ([QRT20, Lemma 3.4 and Corollary 3.5]).

Let $\alpha$ and $\beta$ are quasi-gedodesic rays in $X$, such that $\beta$ is
$(\mathsf{q},\mathsf{Q})$-quasi-geodesic and $\alpha$ is $\kappa$-Morse. If
$\alpha\sim\beta$ then $\beta$ is $\kappa$-Morse with gauge ${\sf
m}_{\alpha}+4{\sf m}_{\alpha}(\mathsf{q},\mathsf{Q})$.

We will reprove this in a greater generality in Proposition 3.1.

Define $\partial_{\kappa}X$ as the set of $\kappa$-Morse quasigeodesic up to
$\sim$. For any $\kappa$-Morse $\beta$ , define $\partial\mathcal{U}(\beta,r)$
as

$\\{\mathbf{a}:\alpha\in\mathbf{a}\text{ is a
$(\mathsf{q},\mathsf{Q})$-quasigeodesic and }r\gg_{\kappa}{\sf
m}_{\beta}(\mathsf{q},\mathsf{Q})\implies\alpha_{\mid
r}\subseteq\mathcal{N}_{\kappa}(\beta,{\sf
m}_{\beta}(\mathsf{q},\mathsf{Q}))\\},$

and then

$\partial\mathcal{B}(\mathbf{b})=\\{\mathcal{V}\subseteq\partial_{\kappa}X:\exists\beta\in\mathbf{b},\exists
r>0,\mathcal{V}\supseteq\partial\mathcal{U}(\beta,r)\\}.$

The topology on $\partial_{\kappa}X$ is defined as the unique one so that the
$\partial\mathcal{B}(\mathbf{b})$ are the neighbourhood systems at
$\mathbf{b}$ [QRT20, Lemma 4.5]; it is metrizable [QRT20, Lemma 4.8]. Finally,
the sublinearly Morse boundary is defined as $\partial
X=\cup_{\kappa}\uparrow\partial_{\kappa}X$.

###  2.5. Sublinear biLipschitz equivalence

In this paragraph, $\theta$ is a sublinear function with the same properties
as $\kappa$.

######  Definition 2.10.

Let $Z$ and $Z^{\prime}$ be two closed unbounded subsets in $X$. Say that $Z$
and $Z^{\prime}$ linearly separate if $d(Z\cap
S(\mathfrak{o},r),Z^{\prime})-r$ stays bounded as $r\to+\infty$.

######  Definition 2.11 ($\theta$-SBE).

Let $(X,\mathfrak{o})$ and $(Y,\mathfrak{o})$ be proper geodesic metric spaces
with basepoints. Let $L\geqslant 1$ be a constant, and let $\theta$ be a
sublinear function as before. We say that $\Phi:X\to Y$ is a
$(L,\theta)$-sublinear biLipschitz equivalence ($\theta$-SBE for short) if

$\frac{1}{L}d(x_{1},x_{2})-\theta(\max(\|x_{1}\|,\|x_{2}\|))\leqslant
d(\Phi(x_{1}),\Phi(x_{2})))\leqslant
Ld(x_{1},x_{2})+\theta(\max(\|x_{1}\|,\|x_{2}\|))$

and $Y=\mathcal{N}_{\theta}(\Phi(X),D)$ for some $D\geqslant 0$.

######  Proposition 2.12 (Inverses).

Let $\Phi:X\to Y$ be a $\theta$-SBE. Then there exists $\overline{\Phi}:Y\to
X$ a $\theta$-SBE and $n>0$ such that for all $x\in X$,

(2.7) $d(x,\overline{\Phi}(\Phi(x)))\leqslant n\cdot\theta(x)$

and for all $y\in Y$,

(2.8) $d(y,{\Phi}(\overline{\Phi}(y)))\leqslant n\cdot\theta(y).$

###### Proof.

This follows from [Cor19, Proposition 2.4], however, let us give here a self-
contained proof. For simplicity, let us assume for now that $\Phi(X)$ is
closed in $Y$; we will see in the end how to remove this assumption if
necessary. For every $y\in Y$, define $\overline{\Phi}(y)$ as some $x\in X$
such that $\Phi(X)$ is a nearest-point projection of $y$ on $\Phi(X)$. By
assumption, $\Phi$ is a $\theta$-SBE, hence applying the last property in
Definition 2.11 one gets that

(2.9) $d(y,\Phi(\overline{\Phi}(y)))\leqslant D\theta(\Phi(x)),$

for some $D\geqslant 0$. This is almost (2.8); let us rework this inequality
slightly. Since we know that $d(y,\Phi(x))\leqslant D\theta(\|y\|)$, when $y$
is far enough from $\mathfrak{o}$, there is some constant $K$ so that as soon
as $\|y\|\geqslant K$, $\|y\|/2\leqslant\|\Phi(x)\|\leqslant 2\|y\|$. It
follows that for some constant $K^{\prime}$, for all $y$,
$\|\Phi(x)\|\leqslant 2\|y\|+K^{\prime}$. Hence

$d(y,{\Phi}(\overline{\Phi}(y)))\leqslant D\theta(\Phi(x))\leqslant
D\theta(2\|y\|+K^{\prime})=O(\theta(y)).$

Thus we proved (2.8). Now for (2.7), note that $x$ and
$\overline{\Phi}(\Phi(x))$ have the same image, namely $\Phi(x)$, through
$\Phi$. So
$\frac{1}{L}d(x,\overline{\Phi}(\Phi(x))-\theta(\max(\|x\|,\|\overline{\Phi}(\Phi(x))\|)\leqslant
d(\Phi(x),\Phi(x))=0$, whence

(2.10) $d(x,\overline{\Phi}(\Phi(x))\leqslant
L\theta(\max(\|x\|,\|\overline{\Phi}(\Phi(x))\|).$

But also
$\frac{1}{L}\|\overline{\Phi}(\Phi(x)\|-\theta(\overline{\Phi}(\Phi(x)))\leqslant\|\Phi(x)\|\leqslant
L\|x\|+\theta(x)$. There exists $K$ such that $\theta(r)\leqslant K+r/(2L))$,
and then

$\frac{1}{2L}\|\overline{\Phi}(\Phi(x)\|\leqslant L\|x\|+\frac{\|x\|}{2L}+2K.$

Plugging this into (2.10),

$d(x,\overline{\Phi}(\Phi(x))\leqslant
L\theta(\max(\|x\|,(2L^{2}+1)\|x\|+2LK)=O(\theta(\|x\|)).$

Finally we need to prove that $\overline{\Phi}$ is a $\theta$-SBE. Applying
the inequality on the right in Definition 2.11 for $\Phi$,

$\displaystyle d(\overline{\Phi}(y),\overline{\Phi}(y^{\prime}))$
$\displaystyle\leqslant
Ld(\Phi\overline{\Phi}(y),\Phi\overline{\Phi}(y^{\prime}))+\theta(\max(\|\overline{\Phi}(y)\|,\|\overline{\Phi}(y^{\prime})\|))$
$\displaystyle\leqslant
Ld(y,y^{\prime})+2\theta(\max(\|y\|,\|y^{\prime}\|,\|\overline{\Phi}(y)\|,\|\overline{\Phi}(y^{\prime})\|))$

Note that

$\frac{1}{L}\|\overline{\Phi}(y)\|-\theta(\|\overline{\Phi}(y)\|)\leqslant
d(\Phi\overline{\Phi}(y),\Phi(\mathfrak{o}))\leqslant\|y\|+O(\theta(\|y\|)).$

so that $\|\overline{\Phi}(y)\|\leqslant 2L\|y\|+M$ for some constant $M$.
Thus

$d(\overline{\Phi}(y),\overline{\Phi}(y^{\prime}))\leqslant
Ld(y,y^{\prime})+2\theta(\max(\|y\|,\|y^{\prime}\|,\|\overline{\Phi}(y)\|,\|\overline{\Phi}(y^{\prime})\|))\leqslant
Ld(y,y^{\prime})+2P\theta(\max(\|y\|,\|y^{\prime}\|)$

for some $P>0$. This proves the inequality on the right in Definition 2.11 for
$\overline{\Phi}$ ; in the exact same way, the left inequality on the left is
obtained by using the inequality on the left for $\Phi$.

It remains to check that
$X=\mathcal{N}_{\theta}(\overline{\Phi}(Y),\overline{D})$ for some
$\overline{D}\geqslant 0$. This follows from (2.7) exactly the same way that
we deduced (2.8) from (2.9).

Finally, $\Phi(X)$ may not be closed in $Y$, but in the construction of
$\overline{\Phi}(y)$, we can relax the condition defining $\overline{\Phi}(y)$
by replacing the nearest-point projection of $y$ with some point at distance
at most $2d(y,\Phi(X))$ from $y$. All the estimates afterwards go through with
additional multiplicative constants. ∎

######  Remark 2.13.

When $\theta=1$, the proof is easier and it is one of the first exercises on
quasiisometries in textbooks; see e.g. [DK18, Exercise 8.12].

######  Lemma 2.14.

Let $\alpha$ be an $(L,\theta)$-ray. Let $\Phi$ and $\overline{\Phi}$ be as in
Proposition 2.12. Let $\widehat{\alpha}$ and
$\widehat{\overline{\Phi}\Phi\alpha}$ be continuous completions of $\alpha$
and $\overline{\Phi}\Phi\alpha$. Then there exists $n$ depending on $L$ and
$\theta$ and $n^{\prime}$ depending on $L,\theta$ and $\Phi$, such that
$\widehat{\alpha}\subset\mathcal{N}_{\theta}(\widehat{\overline{\Phi}\Phi\alpha},n)$
and
$\widehat{\overline{\Phi}\Phi\alpha}\subset\mathcal{N}_{\theta}(\widehat{\alpha},n^{\prime})$.

###### Proof.

By Proposition 2.12, there exists $n_{0}$ such that for every $x$ in $\alpha$,
$d(x,\overline{\Phi}\Phi x)\leqslant n_{0}\theta(x)$. Hence
$\alpha\subset\mathcal{N}_{\theta}(\overline{\Phi}\Phi\alpha,n_{0})$. Morever,
by Equation (2.3) and Remark 2.3, there is $n_{1}$ depending on $L$ and
$\theta$ such that

$d(\widehat{\gamma}(t),\gamma(t)\leqslant
n_{1}\theta(\max(\|\gamma(t)\|,\|\widehat{\gamma}(t)\|)$

for all $t$. It follows that

$\alpha\subset\mathcal{N}_{\theta}(\overline{\Phi}\Phi\alpha,n_{0}+n_{1}).$

By Lemma 2.1, since $d(x,\overline{\Phi}\Phi x)\leqslant n_{0}\kappa(x)$ for
all $x$ in $\alpha$, there are $\mathsf{D}_{1}$, $\mathsf{D}_{2}$ such that
$\mathsf{D}_{1}\kappa(x)\leqslant\kappa(\overline{\Phi}\Phi
x)\leqslant\mathsf{D}_{2}\kappa(x)$ for every $x$ on $\alpha$. Finally,
$\overline{\Phi}\Phi\alpha$ is a $(L^{\prime},\theta)$ ray where $L^{\prime}$
depends on $L$ and $\Phi$. So applying again Equation (2.3) and Remark 2.3,
there exists $n_{1}^{\prime}$ such that

$d(\widehat{\overline{\Phi}\Phi\gamma}(t),\overline{\Phi}\Phi\gamma(t))\leqslant
n_{1}\theta(\max(\|\overline{\Phi}\Phi\gamma(t)\|,\|\widehat{\overline{\Phi}\Phi\gamma}(t)\|).$

Setting $n^{\prime}=n_{0}+n_{1}^{\prime}$ finishes the proof. ∎

There exist several degree of closeness between $\kappa$-rays. The first is
the $\sim$ relation defined earlier.

######  Definition 2.15 ($\kappa$-fellow travelling rays).

Given two rays $\alpha$ and $\beta$ (which are frequently either quasi-
geodesic rays or $\theta$-rays in this paper) we say, $\alpha$ and $\beta$
_$\kappa$ -fellow travel_ each other if there exists $n$ and for all $t>0$, we
have

$d(\alpha(t),\beta(t))\leq n\cdot\kappa(t).$

Further, we say that $\alpha$ and $\beta$ $\kappa$-track each other if there
exists $n_{1}$ such that

$d(\alpha_{r},\beta_{r})\leq n_{1}\cdot\kappa(r).$

for all $r$.

Note that if $\alpha$ and $\beta$ $\kappa$-fellow travel each other, then they
$\kappa$-track each other, however the converse is not true.

######  Lemma 2.16.

Let $\alpha$ and $\alpha^{\prime}$ be fellow-travelling $(L,\theta)$-rays, and
assume that $\alpha$ is $\kappa$-Morse. Then, $\alpha^{\prime}$ is
$2\kappa+\theta$-Morse.

###### Proof.

We will show that the characterization of Definition 2.8 holds here. Let
$\beta$ be quasi-geodesic ray that sublinearly tracks $\alpha^{\prime}$ and
let that sublinear tracking function be $\rho(t)$. Since $\alpha^{\prime}$ and
$\alpha$ are $\kappa$-fellow traveling $\theta$-rays, we have that $\beta$ is
in a $(\rho+\kappa+\theta)$-neighbourhood of $\alpha$. Since $\alpha$ is
$\kappa$-Morse, by Definition 2.8 we have that $\beta$ is in a
$\kappa$-neighbourhood of $\alpha$. But since $\alpha$ is in a $\kappa+\theta$
neighbourhood of $\alpha^{\prime}$, we have that $\beta$ is in a
$2\kappa+\theta$ neighbourhood of $\alpha^{\prime}$. Therefore,
$\alpha^{\prime}$ is $2\kappa+\theta$-Morse. ∎

##  3\. $\theta$-SBE invariance of the $\kappa$-Morse quasi-geodesic rays,
when $\theta=O(\kappa)$

In this section we establish what happens to $\kappa$-Morse quasi-geodesic
rays under $(L,\theta)$ maps. We prove that they behave well in the sense that
they are sent to images sublinearly tracking sublinearly Morse geodesic rays,
provided that $\kappa$ dominates $\theta$.

Precisely, given two sublinear functions $\kappa,\theta$, we say $\kappa$
_dominates_ $\theta$ if there exists constants $C_{1},C_{2}$ and some $t_{0}$
such that for all $t>t_{0}$,

$\kappa(t)\geq C_{1}\theta(t)+C_{2}.$

Therefore in the results proven, we show frequently a ray is
$(\kappa+\theta)$-Morse, which implies it is $\kappa$-Morse if
$\theta\preceq\kappa$ and it is $\theta$-Morse if $\kappa\preceq\theta$.

###  3.1. $\kappa$-Morse rays

Assume that $\alpha$ is a $\kappa$-Morse $(L,\theta)$-ray in a proper metric
space. Then we can establish that a quasi-geodesic ray that tracks $\alpha$
sublinearly is itself $\kappa$-Morse.

######  Proposition 3.1.

Assume that $\kappa$ dominates $\theta$. Let $\alpha$ be an $(L,\theta)$-ray
that is $\kappa$-Morse, and let $\beta\sim\alpha$ be a $({\sf q},{\sf
Q})$-quasigeodesic ray that tracks $\alpha$ sublinearly. Then $\beta$ is
$\kappa$-Morse.

${\mathfrak{o}}$$\alpha$$\beta$$\beta^{\prime}$$\leq\kappa^{\prime}(R)$$\leq\kappa^{\prime}(R+\kappa^{\prime}(R))$$y_{t}$$q_{t}$$s_{t}$$\beta^{\prime}_{R}$$p_{R}$
Figure 2. The setup in the proof of Proposition 3.1.

###### Proof.

The proof is similar to that of [QRT20, Lemma 3.4] where $\beta$ is also
$(\mathsf{q},\mathsf{Q})$-quasi-geodesic but $\alpha$ is a $\kappa$-Morse
quasi-geodesic instead of a $\kappa$-Morse $(L,\theta)$-ray. First, assume
that $\alpha$ is continuous. By an identical proof of [BH99, Lemma III.1.11],
an $(L,\theta)$-ray $\alpha$ can always be made to be continuous as an
$(L,2\theta)$-ray, thus we assume without loss of generality from now on that
$\alpha$ is continuous.

Define $\kappa^{\prime}(r):=d_{X}(\beta_{r},\alpha)$. By Definition 2.5 and
Lemma 2.6, the function $\kappa^{\prime}$ is sublinear. Let us now prove that
$\beta$ is $\kappa$-Morse. Let $r>0$ and let $\beta^{\prime}$ be a $({\sf
q}^{\prime},{\sf Q}^{\prime})$-quasi-geodesic ray such that

$d_{X}(\beta^{\prime}_{R},\beta)\leq\kappa^{\prime}(R)$

for some sufficiently large $R$. Let $p_{R}$ be a nearest point projection of
$\beta^{\prime}_{R}$ to $\beta$; by construction and by triangle inequality,
we have

$\|p_{R}\|\leq 2\|\beta_{R}^{\prime}\|=2R.$

Then, by the triangle inequality,

$\displaystyle d_{X}(\beta^{\prime}_{R},\alpha)$ $\displaystyle\leq
d_{X}(\beta^{\prime}_{R},p_{R})+d_{X}(p_{R},\alpha)$
$\displaystyle\leq\kappa^{\prime}(R)+{\sf m}_{\alpha}({\sf q},{\sf
Q})\cdot\kappa(\|p_{R}\|)\,\,\,\,\text{$\beta$ is a quasi-geodesic ray and
hence}$ is in a $\kappa$-neighbourhood of $\alpha$.
$\displaystyle\leq\kappa^{\prime}(R)+{\sf m}_{\alpha}({\sf q},{\sf
Q})\cdot\kappa(2R).$

Since $\kappa^{\prime\prime}(R):=\kappa^{\prime}(R)+m({\sf q},{\sf
Q})\cdot\kappa(2R)$ is also a sublinear function, and since $\alpha$ is
$\kappa$-Morse, this implies that

(3.1) $\beta^{\prime}|_{r}\subseteq\mathcal{N}_{\kappa}(\alpha,{\sf
m}_{\alpha}({\sf q}^{\prime},{\sf Q}^{\prime})).$

Let $y_{t}$ be any point on $\beta^{\prime}$ with $\|y_{t}\|=t\leq r$. By
construction and triangle inequality, if $q_{t}$ is a nearest point projection
of $y_{t}$ to $\alpha$, we have

$\|q_{t}\|\leq 2\|y_{t}\|=2t.$

Now, if $q$ is any point on $\alpha$ and $s$ is a nearest point projection of
$q$ to $\beta$, by the triangle inequality and the Morse property,

$\|q\|\geq\|s\|-d_{X}(s,q)\geq\|s\|-{\sf m}_{\alpha}({\sf q},{\sf
Q})\cdot\kappa(\|s\|).$

Moreover, again by construction and triangle inequality, $\|s\|\leq 2\|q\|$,
hence by concavity

$d_{X}(s,q)\leq{\sf m}_{\alpha}({\sf q},{\sf Q})\cdot\kappa(\|s\|)\leq 2{\sf
m}_{\alpha}({\sf q},{\sf Q})\cdot\kappa(\|q\|).$

Thus, let $s_{t}$ be a nearest point projection of $q_{t}$ to $\beta$, the
above estimate yields

$\displaystyle d_{X}(q_{t},s_{t})$ $\displaystyle\leq 2{\sf m}_{\alpha}({\sf
q},{\sf Q})\cdot\kappa(\|q_{t}\|)$ $\displaystyle\leq 4{\sf m}_{\alpha}({\sf
q},{\sf Q})\cdot\kappa(t)$

hence, putting everything together,

$\displaystyle d_{X}(y_{t},\beta)$ $\displaystyle\leq
d_{X}(y_{t},q_{t})+d_{X}(q_{t},s_{t})$ $\displaystyle\leq{\sf m}_{\alpha}({\sf
q}^{\prime},{\sf Q}^{\prime})\cdot\kappa(t)+4{\sf m}_{\alpha}({\sf q},{\sf
Q})\cdot\kappa(t)$

which, by setting ${\sf m}_{\beta}({\sf q}^{\prime},{\sf Q}^{\prime}):={\sf
m}_{\alpha}({\sf q}^{\prime},{\sf Q}^{\prime})+4{\sf m}_{\alpha}({\sf q},{\sf
Q})$, proves the claim. Finally, $\alpha$ may not be continuous. In the case
where it is not, using the notation from Lemma 2.4, we have to check that
$\widehat{\alpha}$ inherits the $\kappa$-Morse property, which follows readily
from Lemma 2.16. ∎

Building on these results we are now ready to establish the map on
${\partial_{\kappa}}X$ that is induced by an SBE.

######  Lemma 3.2.

If $\alpha$ is a (q, Q)-$\kappa$-Morse quasi-geodesic ray in $X$, and $\Phi$
is an $(L,\theta)$-sublinear bi-Lipschitz equivalence between two proper
metric spaces $X$ and $Y$. Then $\overline{\Phi}\Phi\alpha$ is
$(\kappa+\theta)$-Morse.

###### Proof.

Consider $\alpha,\overline{\Phi}\Phi\alpha$. By Proposition 2.12, every point
$x$ of $\alpha$ is $\theta(x)$ away from the point $\overline{\Phi}\Phi(x)$.
Consider a $({\sf q},{\sf Q})$-quasi-geodesic $\gamma$ that is sublinearly
tracking $\overline{\Phi}\Phi\alpha$. Let this tracking function be denoted
$\kappa^{\prime}$.

Then, there exists a constant $C$ such that any point $y$ on $\gamma$ is
distance at most $C(\kappa^{\prime}(y)+\theta(y))$ from $\alpha$. Indeed,
$d(y,\overline{\Phi}\Phi\alpha_{\|y\|})\leqslant\kappa^{\prime}(y)$ and then
by Proposition 2.12,
$d(y,\alpha)\leqslant\kappa^{\prime}(y)+\widehat{\theta}(y)$ where
$\widehat{\theta}\leqslant C\theta$ for some $C$ by (2.2).

Since $\alpha$ is $\kappa$-Morse and $C(\kappa^{\prime}(y)+\theta(y))$ is a
sublinear function, by Proposition 3.1 we have that $\gamma$ is in a
$\kappa$-neighbourhod of $\alpha$ and $\gamma$ is $\kappa$-Morse.

Therefore we have that $\gamma$ is in $\mathcal{N}_{\kappa}(\alpha,m)$ for
some $m$. Since $\alpha$ is $\theta$-tracking $\overline{\Phi}\Phi\alpha$, we
have that $\gamma$ is $m\cdot\kappa+\theta$ tracking
$\overline{\Phi}\Phi\alpha$. Thus we have shown that any quasi-geodesic ray
that sublinearly tracks $\overline{\Phi}\Phi\alpha$ with rate
$m\cdot\kappa+\theta$. Thus $\overline{\Phi}\Phi\alpha$ is
$(\kappa+\theta)$-Morse. ∎

$\alpha$$\gamma$$\overline{\Phi}\Phi\alpha$ Figure 3. Proof of Lemma 3.2.

######  Proposition 3.3.

If $\alpha$ is a (q, Q) $\kappa$-Morse quasi-geodesic ray in $X$, and $\Phi$
is an $(L,\theta)$-sublinear bi-Lipschitz equivalence between two proper
metric spaces $X$ and $Y$. Then $\Phi\alpha$ is $(\kappa+\theta)$-Morse.

###### Proof.

Consider a $({\sf q},{\sf Q})$-quasi-geodesic ray $\gamma$ that sublinearly
tracks $\Phi\alpha$ and let the tracking function be $\kappa^{\prime}$.
Consider the image $\overline{\Phi}\gamma$. $\overline{\Phi}\gamma$ is
distance $\theta+\kappa^{\prime}$ from $\overline{\Phi}\Phi\alpha$. Since
$\overline{\Phi}\Phi\alpha$ is $\theta$-tracking $\alpha$ we have that
$\overline{\Phi}\gamma$ is $2\theta+\kappa^{\prime}$-tracking $\alpha$. By
Proposition 3.1, $\overline{\Phi}\gamma$ is $\kappa$-Morse. Therefore, since
$\alpha$ is a quasi-geodesic that sublinearly tracks $\overline{\Phi}\gamma$,
we have that $\alpha$ and $\overline{\Phi}\gamma$ are $\kappa$-close.

Now apply $\Phi$ to both $\alpha$ and $\overline{\Phi}\gamma$. $\Phi\alpha$
and $\Phi\overline{\Phi}\gamma$ are $\kappa+\theta$ apart. Since
$\Phi\overline{\Phi}\gamma$ and $\gamma$ are $\theta$ apart, then we have that
$\Phi\alpha$ and $\gamma$ are at most $\kappa+2\theta$ apart. This holds for
every $\gamma$ and thus we have that $\Phi\alpha$ is $(\kappa+\theta)$-Morse.
∎

These two results guarantee that under a sublinear biLipschitz equivalence,
the image of a $\kappa$-Morse quasi-geodesic ray is an $(L,\theta)$-ray that
carries the $\kappa$-Morse property, i.e. any quasi-geodesic ray that
sublinearly tracks this set tracks it with a uniformly controlled sublinear
function. The next two results shows that the latter set, i.e. a
$\kappa$-Morse $(L,\theta)$-ray still has strong association with a geodesic
ray and thus can be connected with a $\kappa$-Morse equivalence class in
${\partial_{\kappa}}X$ in Section 4.

######  Lemma 3.4.

Let $\alpha$ be a $\kappa$-Morse $(L,\theta)$-ray. Then there exists a
geodesic ray $\underline{a}$ such that $\underline{a}$ and $\alpha$
$\kappa$-track each other.

$\alpha$$\underline{a}$$\alpha(i_{1})$$\alpha(i_{2})$$\alpha(i_{3})$ Figure 4.
Proof of Lemma 3.4.

###### Proof.

Since $\alpha$ is $\kappa$-Morse, let ${\sf m}_{\alpha}({\sf q},{\sf Q})$ be
its $\kappa$-Morse gauges. Consider the geodesic segments connecting the base-
point and $\alpha(i)$, $i=1,2,3\ldots$.

Since $\alpha$ is $\kappa$-Morse, for every $r>0$, there exists $i_{r}$ such
that

$\alpha(i_{r})\in\mathcal{N}_{1}(\alpha,1)\,\Longrightarrow\,[{\mathfrak{o}},\alpha(r)]\in\mathcal{N}_{\kappa}(\alpha,m(1,0)).$

The space $X$ is assumed to be proper. Thus by the Arzelá-Ascoli Theorem, for
some extraction $\\{i_{n}\\}$, the subsequence
$\\{[{\mathfrak{o}},\alpha(i_{n})],n=1,2,3\ldots\\}$ converges to a geodesic
ray $\underline{a}$.

For each $n$, all but the first $n-1$ segments in this sequence has the
property that the intersection of them with the ball of radius $i_{n}$ is in
the neighbourhood $\mathcal{N}_{\kappa}(\alpha,m(1,0))$, thus their limit
$\underline{a}$ is in the neighbourhood $\mathcal{N}_{\kappa}(\alpha,m(1,0))$.
Thus $\underline{a}$ and $\alpha$ $\kappa$-track each other. ∎

Note that we used the $\kappa$-Morse property in an essential way.

When $X$ is Gromov hyperbolic, Lemma 3.4 is a consequence of [Pal20, Lemma
3.4].

######  Proposition 3.5.

If $\alpha$ is an $(L,\theta)$-ray that is $\kappa$-Morse, then $\alpha$ is in
a $\kappa$-neighbourhood of a unique $\kappa$-Morse geodesic ray, up to
sublinear tracking. Thus a $\kappa$-Morse $(L,\theta)$-ray is associated with
a unique equivalence class $\textbf{a}\in{\partial_{\kappa}}X$.

###### Proof.

By Lemma 3.4, there exists a geodesic ray $\underline{a}$ such that
$\underline{a}$ and $\alpha$ $\kappa$-tracks each other. Then Proposition 3.1
implies that $\underline{a}$ is $\kappa$-Morse. Since every $\kappa$-Morse
geodesic ray is in a unique equivalence class
$\textbf{a}\in{\partial_{\kappa}}X$, we get that $\alpha$ is associated with
$\textbf{a}\in{\partial_{\kappa}}X$. Suppose $\alpha$ is in a
$\kappa$-neighbourhood of another $\kappa$-Morse geodesic ray
$\underline{a}^{\prime}$, we have that $\underline{a}^{\prime}$ and
$\underline{a}$ $\kappa$-track each other and thus
$\underline{a}^{\prime}\in\textbf{a}$. ∎

##  4\. SBE on sublinearly Morse boundaries

###  4.1. Proof of Theorem A

In this section we look at the induced map of a given SBE $\Phi\colon X\to Y$
on ${\partial_{\kappa}}X$. To begin with, we need the following basic
observation about $(L,\theta)$-ray and geodesics that are in sublinear
neighbourhoods of each other. The proof is identical to the analogous claim in
[QRT20] regarding quasi-geodesic rays and geodesic rays:

######  Lemma 4.1.

Let $\beta$ be a $(L,\theta)$-ray and $\underline{a}$ be a geodesic ray, both
based at ${\mathfrak{o}}\in X$. Suppose that

$\beta\subseteq\mathcal{N}_{\kappa}(\underline{a},m)$

for some function $\kappa$ and some constant $m$. Then we also have

$\underline{a}\subseteq\mathcal{N}_{\kappa+\theta}(\beta,m^{\prime})$

for some constant $m^{\prime}$. Especially, if $\kappa$ dominates $\theta$,
then for some $m^{\prime}$

$\underline{a}\subseteq\mathcal{N}_{\kappa}(\beta,m^{\prime}).$

${\mathfrak{o}}$$\underline{a}$$\widehat{\beta}$$y$$q$$z$ Figure 5. $\lVert
y\rVert=\lVert z\rVert$ and $q\in\pi_{\underline{a}}(z)$ as in the proof of
Lemma 4.1.

###### Proof.

The core of the proof is the same as that of [QRT20, Lemma 3.1]; we reproduce
it here for completeness. First, assume for simplicity that $\beta$ is a
continuous $\theta$-ray. Let $y\in\underline{\alpha}$ be a point and let
$r:=\lVert y\rVert$. Let $z\in\beta$ be a point such that $\|z\|=r$ and let
$q$ be a nearest point projection of $z$ to $\underline{a}$. By assumption,

$d_{X}(z,q)\leq m\cdot\kappa(r).$

On the other hand,

$\displaystyle d_{X}(y,q)$ $\displaystyle=|\|y\|-\lVert q\rVert|$ since
$\underline{a}$ is geodesic $\displaystyle=|\|z\|-\lVert q\rVert|$
$\displaystyle\leq d_{X}(z,q)$ $\displaystyle\text{by the triangle
inequality}.$

Therefore we have

$d_{X}(y,\beta)\leq d_{X}(y,z)\leq d_{X}(y,q)+d_{X}(q,z)\leq 2d_{X}(z,q)\leq
2m\cdot\kappa(r)$

which completes the proof in this case. Note that we can take $m^{\prime}=2m$.
Now, $\beta$ may not be continuous. Nevertheless, by Lemma 2.4 there is a
continuous $\theta$-ray $\widehat{\beta}$ that $\theta$-fellow travels with
$\beta$. Keeping $y$ as before, and applying the previous argument to
$\widehat{\beta}$, there is $z\in\widehat{\beta}$ with $d(y,z)\leqslant
2m\cdot\kappa(r)$; but then also, $d(z,\beta)\leqslant m_{1}\theta(r)$,and
then $d(y,\beta)\leqslant 2m\kappa(r)+m_{1}\theta(r)\leqslant
m^{\prime}(\theta(r)+\kappa(r))$, as required. ∎

Let $\Phi\colon X\to Y$ be a $(L,\theta)$-SBE; we are now ready to define the
induced map $\Phi_{\star}$.

Given a $(\mathsf{q},\mathsf{Q})$-quasi-geodesic ray
$\zeta\colon\thinspace[0,\infty)\to X$ in $X$, we assume without loss of
generality that the image of $\zeta$ is a continuous path. Let $\Phi\zeta$ be
the $(L\mathsf{q},\theta)$-ray in $Y$ constructed from the composition of
$\zeta$ and $\Phi$.

######  Proposition 4.2.

Assume that $\kappa$ dominates $\theta$, and let $\Phi$ be a
$(L,\theta)$-sublinear bi-Lipschitz equivalence from $X$ to $Y$, where $X$ and
$Y$ are proper geodesic metric spaces. Two $\kappa$-Morse quasi-geodesics
$\alpha$ and $\beta$ in $X$ $\kappa$-fellow travel each other if and only if
$\Phi\alpha$ and $\Phi\beta$ $\kappa$-fellow travel each other in $Y$.

###### Proof.

If $\alpha$ and $\beta$ in $X$ $\kappa$-fellow travel each other with
multiplicative constant $n$, then at radius $r$, the distances between points
of $\Phi\alpha$ and $\Phi\beta$ is apart by

$L(n\cdot\kappa(r))+\theta(r)\leq(Ln+1)\kappa(r)$

thus $\Phi\alpha$ and $\Phi\beta$ $\kappa$-fellow travel each other in $Y$. By
Proposition 2.12, there exists an inverse, $\overline{\Phi}$, that is also an
$(L,\theta)$-SBE. Thus we have that if $\Phi\alpha$ and $\Phi\beta$ are
$n^{\prime}\cdot\kappa(r)$-tracking, then $\overline{\Phi}\Phi\alpha$ and
$\overline{\Phi}\Phi\beta$ are $(Ln^{\prime}+1)\kappa(r)$-tracking each other
by the preceding argument. Furthermore, $\alpha,\beta$ are $\theta(r)$
tracking $\overline{\Phi}\Phi\alpha$ and $\overline{\Phi}\Phi\beta$,
respectively. Thus $\alpha,\beta$ are tracking each other with distance at
most

$(Ln^{\prime}+1)\kappa(r)+\theta(r)+\theta(r)\leq(Ln^{\prime}+3)\kappa(r).\qed$

######  Definition 4.3.

It follows from Proposition 4.2 that two quasi-geodesics $\zeta$ and $\xi$ in
$X$ $\kappa$–fellow travel each other if and only if $\Phi\zeta$ and $\Phi\xi$
$\kappa$-fellow travel each other in $Y$. Also by Corollary 3.3, the property
of being in Morse-neighbourhood of a Morse geodesic ray is preserved under an
$(L,\theta)$-SBE. Hence, $[\zeta]\in\partial_{\kappa}X$ if and only if
$[\Phi\zeta]\in\partial_{\kappa}Y$. We write $\textbf{a}=[\zeta]$ and
$\textbf{b}=[\Phi\zeta]$ and we thus define

$\Phi_{\star}(\textbf{a})=\textbf{b}.$

######  Theorem 4.4.

Consider proper geodesic metric spaces $X$ and $Y$, let $\Phi\colon\thinspace
X\to Y$ be a $(L,\theta)$-sublinear bi-Lipschitz equivalence between $X$ and
$Y$ and let $\overline{\Phi}$ denote its inverse as in Proposition 2.12. Let
$\Phi_{\star}$ be defined by Definition 4.3. Then for every sublinear function
$\kappa$ that dominates $\theta$,
$\Phi_{\star}\colon\thinspace\partial_{\kappa}X\to\partial_{\kappa}Y$ is a
homeomorphism.

###### Proof.

By Definition 4.3, $\Phi_{\star}$ defined as above gives a bijection between
$\partial_{\kappa}X$ and $\partial_{\kappa}Y$. We need to show that
$(\Phi_{\star})^{-1}$ is continuous. Then, the same argument applied in the
other direction will show that $\Phi_{\star}$ is also continuous which means
$\Phi_{\star}$ is a homeomorphism.

Let $\mathcal{V}$ be an open set in ${\partial_{\kappa}}X$,
$\textbf{b}_{X}\in\mathcal{V}$ and $\mathcal{U}_{\kappa}(\textbf{b}_{X},{\sf
r})$ be a neighbourhood of $\textbf{b}_{X}$ that is contained in
$\mathcal{V}$. Let $\textbf{b}_{Y}=\Phi_{\star}(\textbf{b}_{X})$. We need to
show that there is a constant ${\sf r}^{\prime}$ such that, for every point
$\textbf{a}_{Y}\in\mathcal{U}_{\kappa}(\textbf{b}_{Y},{\sf r}^{\prime})$, we
have

$(\Phi_{\star})^{-1}(\textbf{a}_{Y})=\textbf{a}_{X}\in\mathcal{U}_{\kappa}(\textbf{b}_{X},{\sf
r}).$

$\mathcal{V}$$\mathcal{U}_{\kappa}(\mathbf{b}_{X},\mathsf{r})$$\mathcal{U}_{\kappa}(\mathbf{b}_{Y},\mathsf{r}^{\prime})$$\mathbf{b}_{X}$$\mathbf{a}_{X}$$\mathbf{b}_{Y}$$\mathbf{a}_{Y}$$\Phi_{\star}$
Figure 6. $\Phi_{\star}$ is a homeomorphism.

By Proposition 3.5, if $\zeta$ is a $({\sf q},{\sf Q})$–quasi-geodesic ray in
$X$ then $\Phi\zeta$ is in a $\kappa$-neighbourhood of a unique $\kappa$-Morse
geodesic ray, up to sublinear fellow travelling. Fix a particular geodesic
representative $c$ from that class $[\Phi\zeta]$ and let $({\sf
q}^{\prime},{\sf Q}^{\prime})$ be the smallest pair of constants such that

$\Phi\zeta\in\mathcal{N}_{\kappa}(c,{\sf m}_{b_{Y}}({\sf q}^{\prime},{\sf
Q}^{\prime})).$

By Proposition 3.3 ${\sf m}_{b_{Y}}({\sf q}^{\prime},{\sf Q}^{\prime})$ only
depends on $q,Q,L,\theta$ and $\zeta$. Thus we can consider a function
$\Phi^{\prime}\colon\thinspace\mathbb{R}^{2}\to\mathbb{R}^{2}$ that takes (q,
Q) to $({\sf q}^{\prime},{\sf Q}^{\prime})$ by the map $\Phi$ on
$\kappa$-Morse quasi-geodesic rays. Roughly speaking, $\Phi\zeta$ "can be
thought of" as a $({\sf q}^{\prime},{\sf Q}^{\prime})$-quasi-geodesic ray in
terms of their sublinear deviation from the geodesic representative.

Next, let $b_{Y}$ be the representative geodesic ray in $\textbf{b}_{Y}$ and
let ${\sf m}_{b_{X}}$ and ${\sf m}_{b_{Y}}$ be their $\kappa$-Morse gauges
respectively. By Lemma 4.1, there is a constant ${\sf n}_{1}$ depending on
$L,\kappa$ such that

$\Phi b_{X}\subset\mathcal{N}_{\kappa}(b_{Y},{\sf n}_{1}).$

Next, we let

${\sf n}=L\big{(}{\sf m}_{b_{Y}}({\sf q}^{\prime},{\sf Q}^{\prime})+{\sf
n}_{1}\big{)}(L+1)+1$

and let ${\sf R}={\sf R}(b_{X},{\sf r},{\sf n},\kappa)$ as in Definition 2.7.
Choose ${\sf r}^{\prime}$ such that ${\sf r}^{\prime}\geq L\,{\sf
R}+\kappa(r)$. Now let $\alpha\in\textbf{a}_{X}$ be a $({\sf q},{\sf
Q})$–quasi-geodesic in $X$, where

  1. (1)

$\Phi\alpha$ is a ray that is associated with an element in
${\partial_{\kappa}}Y$ that is in $\mathcal{U}_{\kappa}(\textbf{b}_{Y},{\sf
r}^{\prime})$, and

  2. (2)

${\sf q},{\sf Q}$ is such that ${\sf m}_{b_{X}}({\sf q},{\sf Q})$ is small
compared to ${\sf r}$.

Pick $x\in\alpha_{X}|_{{\sf R}}$. Then $\Phi x\in\Phi\alpha|_{{\sf
r}^{\prime}}$ and we have

$\displaystyle d_{X}(x,b_{X})$ $\displaystyle\leq L(d_{Y}(\Phi(x),\Phi
b_{X})+\theta(x)$ $\displaystyle\leq
L\Big{(}d_{Y}(\Phi(x),b_{Y})+d(\Phi(x)_{b_{Y}},b_{X})\Big{)}+\theta(x)$
$\displaystyle\leq L\Big{(}d_{Y}(\Phi(x),b_{Y})+{\sf n}_{1}\cdot\kappa(\Phi
x)\Big{)}+\theta(x)$ $\displaystyle\leq L\big{(}{\sf m}_{b_{Y}}({\sf
q}^{\prime},{\sf Q}^{\prime})+{\sf n}_{1}\big{)}\cdot\kappa(\Phi x)+\theta(x)$
$\displaystyle\leq L\big{(}{\sf m}_{b_{Y}}({\sf q}^{\prime},{\sf
Q}^{\prime})+{\sf n}_{1}\big{)}\cdot\kappa(\Phi x)+\kappa(x)$

We also have

$\kappa(\Phi x)\leq L\kappa(x)+\theta(x)\leq(L+1)\kappa(x)$

Combine the preceding inequalities we have

$\displaystyle d_{X}(x,b_{X})$ $\displaystyle\leq L\big{(}{\sf m}_{b_{Y}}({\sf
q}^{\prime},{\sf Q}^{\prime})+{\sf n}_{1}\big{)}\cdot(L+1)\kappa(x)+\kappa(x)$
$\displaystyle\leq\Big{(}L\big{(}{\sf m}_{b_{Y}}({\sf q}^{\prime},{\sf
Q}^{\prime})+{\sf n}_{1}\big{)}(L+1)+1\Big{)}\kappa(x)$

imply that

$\alpha|_{{\sf R}}\subset\mathcal{N}_{\kappa}(b_{X},{\sf n}).$

Now, Definition 2.7 implies that

$\alpha|_{\sf r}\subset\mathcal{N}_{\kappa}(b_{X},{\sf m}_{b_{X}}).$

Therefore, $\textbf{a}_{X}\in\mathcal{U}_{\kappa}(\textbf{b}_{X},{\sf r})$ and

$(\Phi_{\star})^{-1}\mathcal{U}_{\kappa}(\textbf{b}_{Y},{\sf
r}^{\prime})\subset\mathcal{U}_{\kappa}(\textbf{b}_{X},{\sf r}).$

But $\mathcal{U}_{\kappa}(\textbf{b}_{Y},{\sf r}^{\prime})$ contains an open
neighbourhood of $\textbf{b}_{Y}$, therefore, $\textbf{b}_{Y}$ is in the
interior of $\Phi\mathcal{V}$. This finishes the proof. ∎

###  4.2. An application

In [Be19], Behrstock considers an infinite family of right-angled Coxeter
groups. The definining graphs for one of the smallest pair of Behrstock's
groups are as follows. $\Gamma_{14}$ is the simple graph on $28$ vertices
$\\{a_{1},a_{2},\ldots,\,a_{14},b_{14}\\}$ where for
$c,c^{\prime}\in\\{a,b\\}$ there is an edge between $c_{i}$ and
$c^{\prime}_{j}$ if and only if $|i-j|=1$. $\Gamma$ is the graph $\Gamma_{14}$
to which we add a $5$-cycle $a_{1}-a_{4}-a_{7}-a_{10}-a_{13}$.

$a_{1}$$b_{1}$$a_{2}$$b_{2}$$a_{3}$$b_{3}$$a_{4}$$b_{4}$$a_{5}$$b_{5}$$a_{6}$$b_{6}$$a_{7}$$b_{7}$$a_{8}$$b_{8}$$a_{9}$$b_{9}$$a_{10}$$b_{10}$$a_{11}$$b_{11}$$a_{12}$$b_{12}$$a_{13}$$b_{13}$
Figure 7. The graph $W$ (when compared to $W_{13}$, additional edges are
colored).

Let $W_{13}$ and $W$ be the right-angled Coxeter groups with defining graphs
$\Gamma_{13}$ and $\Gamma^{\prime}$ as above respectively.

######  Lemma 4.5.

$W$ and $W_{13}$ have quadratic divergence. In particular, they are not
hyperbolic relative to any of their subgroups of infinite index, and their
asymptotic cones are not tree-graded spaces.

The fact that $W$ has quadratic divergence and is not relatively hyperbolic
was checked by Behrstock [Be19]. We follow the same method, and apply it to
$W_{13}$ as well below.

###### Proof.

Let us prove that $W$ and $W_{13}$ have the Dani-Thomas $\mathcal{CFS}$
property, defined in [DT15, 4.2]. Dani and Thomas associate new graphs to the
defining graphs of right-angled Coxeter groups. Let us denote $\Lambda$ and
$\Lambda_{13}$ the graphs associated to $\Gamma$ and $\Gamma_{13}$
respectively. The vertex sets in $\Lambda$ and in $\Lambda_{13}$ are the four-
cycles in $\Gamma$ and $\Gamma_{13}$ respectively, and there is an edge
between $\lambda$ and $\lambda^{\prime}$ if $\lambda$ and $\lambda^{\prime}$
share a common edge. In the graph $\Lambda$, there are $56$ vertices,
described in terms of loops in $\lambda$ as follows:

$\displaystyle\lambda_{i}^{a}=a_{i}-a_{i+1}-a_{i+2}-b_{i+1}$ $\displaystyle
i=1,\ldots,11$ $\displaystyle\lambda_{i}^{b}=b_{i}-b_{i+1}-b_{i+2}-a_{i+1}$
$\displaystyle i=1,\ldots,11$
$\displaystyle{\lambda^{\prime}}_{i}^{a}=a_{i}-a_{i+1}-b_{i+2}-b_{i+1}$
$\displaystyle i=1,\ldots,11$
$\displaystyle{\lambda^{\prime}}_{i}^{b}=b_{i}-b_{i+1}-a_{i+2}-a_{i+1}$
$\displaystyle i=1,\ldots,11.$
$\displaystyle\mu_{i}=a_{i}-a_{i+1}-b_{i}-b_{i+1}$ $\displaystyle
i=1,\ldots,12$

There is an edge $\lambda_{i}^{a}$ and ${\lambda^{\prime}}_{j}^{a}$ and
between $\lambda_{i}^{b}$ and ${\lambda^{\prime}}_{j}^{b}$ if and only if
$0\leqslant j-i\leqslant 1$, an edge between ${\lambda^{\prime}}_{i}^{a}$ and
${\lambda^{\prime}}_{j}^{b}$ if and only if $|j-i|=1$, an edge between
$\lambda_{i}^{a}$ and ${\lambda^{\prime}}_{j}^{b}$ and between
$\lambda_{i}^{b}$ and ${\lambda^{\prime}}_{j}^{a}$ if and only if $0\leqslant
i-j\leqslant 1$, and, finally, an edge between $\mu_{i}$ and $\lambda^{a}_{j}$
or $\lambda^{b}_{j}$ if $0\leqslant i-j\leqslant 1$. It is clear from this
that $\Lambda$ is connected. Moreover, any vertex of $\Gamma$ is present in at
least one loop that is represented by a vertex in $\Lambda$, that is, using
the words of Dani-Thomas, the single component of $\Lambda$ has full support.
This is precisely the property $\mathcal{CFS}$.

It now follows from [DT15] that the groups $W$ and $W_{13}$ have quadratic
divergence. Since the relatively hyperbolic groups have exponential divergence
$W$ and $W_{13}$ are not relatively hyperbolic [Sis12], and their asymptotic
cones are not tree-graded spaces [DS].

Let's now consider the graph $\Lambda_{13}$. As compared to $\Lambda$, there
are sixteen additional loops, namely

$\displaystyle a_{i}-a_{i+3}-a_{i+2}-a_{i+1}$ $\displaystyle
a_{i}-a_{i+3}-b_{i+2}-b_{i+1}$ $\displaystyle a_{i}-a_{i+3}-b_{i+2}-a_{i+1}$
$\displaystyle a_{i}-a_{i+3}-a_{i+2}-b_{i+1}$

for $i\in\\{1,4,7,10\\}$. All these new loops are easily seen to be connected
in $\Lambda_{13}$ to one of the $\lambda_{i}^{a}$, $\lambda_{i}^{b}$,
${\lambda^{\prime}}_{i}^{a}$, ${\lambda^{\prime}}_{i}^{b}$. It follows that
$W_{13}$ has the $\mathcal{CFS}$ property. ∎

######  Lemma 4.6.

Let $\kappa$ be a sublinear function, ${\partial_{\kappa}}W_{13}$ is totally
disconnected.

###### Proof.

Let $c_{i}=a_{i}b_{i}$, then it is known that the following graph defining a
right-angled Artin group on the vertices embeds in $W_{13}$ as a finite index
subgroup [DJ00].

$c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{5}$$c_{6}$$c_{7}$$c_{8}$$c_{9}$$c_{10}$$c_{11}$$c_{12}$$c_{13}$
Figure 8. Right-angled Artin group defined by a path, $A_{13}$.

By [IZ, Theorem 1.1], ${\partial_{\kappa}}A_{13}$ continuously injects into
the Gromov boundary of the contact graph of the associated Salvetti complex.
Since contact graphs are quasi-isometric to trees [Hag14], their Gromov
boundaries are totally disconnected. Thus ${\partial_{\kappa}}A_{13}$ is
totally disconnected. Since $W_{13}$ is quasi-isometric to $A_{13}$, we have
that ${\partial_{\kappa}}W_{13}$ is also totally disconnected. ∎

######  Proposition 4.7.

$W$ and $W_{13}$ are not sublinear biLipschitz equivalent.

###### Proof.

The $5$-cycle in $\Gamma$ induces an embedding of a finite-index subgroup of
the Coxeter group $\Delta=\langle r,s,t\mid(rs)^{5},(st)^{2},(tr)^{4}\rangle$
in $W$. $\Delta$ itself is a finite-index subgroup of a hyperbolic surface
group. Behrstock proves that the embedding in $W$ is stable. Since $\Delta$ is
a finite index subgroup of a hyperbolic group, the sublinearly Morse boundary
of $\Delta$ is also a circle. On the other hand, by Lemma 4.6,
${\partial_{\kappa}}W_{13}$ is totally disconnected and thus it does not
contain a circle. Therefore ${\partial_{\kappa}}W$ and
${\partial_{\kappa}}W_{13}$ are not homeomorphic, and thus we conclude from
Theorem 4.4 that $W$ and $W_{13}$ are not sublinear biLipschitz equivalent. ∎

It follows from the work of Hagen, Kazachkov and Casals-Ruiz [HKR] that $W$
and $W_{13}$ have unique asymptotic cones. Up to our knowledge, it was not
known whether the asymptotic cones of $W_{13}$ and $W$ are biLipschitz
homeomorphic or not.

######  Remark 4.8.

It is also worth noting that even though we distinguish the $\kappa$-Morse
boundaries of $W_{13}$ and $W$ by the Morse boundaries that are proper
subspaces of their $\kappa$-Morse boundaries, Morse boundaries in general are
not SBE-invariant, therefore it was necessary to invoke Theorem 4.4.

##  5\. Random walk on groups and sublinear rays

### Random walks

Let $G$ be a countable group, and let $\mu$ be a probability measure on a
symmetric generating set of $G$. We consider the _step space_
$(G^{\mathbb{N}},\mu^{\mathbb{N}})$, whose elements we denote as $(g_{n})$.
The _random walk driven by_ $\mu$ is the $G$-valued stochastic process
$(w_{n})$, where for each $n$ we define the product

$w_{n}:=g_{1}g_{2}\dots g_{n}.$

We denote as $(\Omega,\mathbb{P})$ the _path space_ , i.e. the space of
sequences $(w_{n})$, where $\mathbb{P}$ is the measure induced by pushing
forward the measure $\mu^{\mathbb{N}}$ from the step space. Elements of
$\Omega$ are called _sample paths_ and will be also denoted as $\omega$.
Finally, let $T:\Omega\to\Omega$ be the left shift on the path space.

### Background on boundaries

Let us recall some fundamental definitions from the boundary theory of random
walks. For a more extensive exposition, see [Kai00]. Let $(B,\mathcal{A})$ be
a measurable space on which $G$ acts by measurable isomorphisms; a measure
$\nu$ on $B$ is $\mu$-_stationary_ if $\nu=\int_{G}g_{\star}\nu\ d\mu(g)$, and
in that case the pair $(B,\nu)$ is called a $(G,\mu)$-_space_. Recall that a
_$\mu$ -boundary_ is a measurable $(G,\mu)$-space $(B,\nu)$ such that there
exists a $T$-invariant, measurable map
$\textbf{bnd}:(\Omega,\mathbb{P})\to(B,\nu)$, called the _boundary map_.

Moreover, a function $f:G\to\mathbb{R}$ is $\mu$-_harmonic_ if
$f(g)=\int_{G}f(gh)\ d\mu(h)$ for any $g\in G$. We denote by
$H^{\infty}(G,\mu)$ the space of bounded, $\mu$-harmonic functions. One says a
$\mu$-boundary is the _Poisson boundary_ of $(G,\mu)$ if the map

$\Psi:H^{\infty}(G,\mu)\to L^{\infty}(B,\nu)$

given by $\Psi(f)(g):=\int_{B}f\ dg_{\star}\nu$ is a bijection. The Poisson
boundary $(B,\nu)$ is the maximal $\mu$-boundary, in the sense that for any
other $\mu$-boundary $(B^{\prime},\nu^{\prime})$ there exists a
$G$-equivariant, measurable map $p:(B,\nu)\to(B^{\prime},\nu^{\prime})$. The
result of this section concerns the shape of all sample paths.

######  Theorem 5.1.

Let $G$ be the mapping class group $Mod(S)$ of a finite type surface, or let
$G$ be a relatively hylic group. Let $\mu$ be a probability measure on G with
finite first moment with respect to the metric $d$, such that the semigroup
generated by the support of $\mu$ is a non-amenable group. Let
$\kappa(r)=\log(2+r)$. Then there exists a constant $A$ such that almost every
sample path $(w_{n})$ is such that $(w_{n}{\mathfrak{o}})$ is a
$\kappa$-sublinear ray. Moreover, the upper and lower large-scale Lipschitz
constants of $(w_{n})$ are equal, that is, there exists $A$ such that

(5.1) $A|n-m|-\kappa(\max(n,m))\leqslant d(w_{n},w_{m})\leqslant
A|n-m|+\kappa(\max(n,m)).$

###### Proof.

By Theorem $C$ in [QRT20],

$\limsup_{n\to\infty}\frac{d_{w}(w_{n},\gamma_{\omega})}{\log n}<+\infty.$

Thus there exists a $C$ such that

$\limsup_{n\to\infty}d_{w}(w_{n},\gamma_{\omega})\leq C\log n$

By [MT18], weakly hyperbolic groups have positive drift on their respective
associated hyperbolic spaces. And by the Distance Formula (Masur-Minsky
[MM00]), distance (to the origin) in the random walk on the group is coarsely
bounded by the distance to the origin in the associated curve graph $d_{S}$.
Thus random walks on a mapping class groups have positive drifts ([MT18]) That
is to say, there exists an $A$ such that for $n$ large enough

$d(w_{n}{\mathfrak{o}},{\mathfrak{o}})\geq An.$

Let $p_{n}$ denote the nearest point projection of $w_{n}$ to
$\gamma_{\omega}$. We have by triangle inequality

$d({\mathfrak{o}},p_{n})\geq
d(w_{n}{\mathfrak{o}},{\mathfrak{o}})-d(w_{n}{\mathfrak{o}},p_{n}))\geq
An-C\log n.$

We have that as $n\to\infty$

$An\geq\lim d({\mathfrak{o}},p_{n})\lim\geq An-C\log
n=An=d({\mathfrak{o}},\gamma(An)),$

where the last equality comes from the fact that $\gamma$ is unit speed.
Therefore

$d(w_{n}{\mathfrak{o}},p_{n})=C\log n=d(w_{n}{\mathfrak{o}},\gamma(An)).$

Next we construct a map on $\gamma$ such that for each $i\in\mathbb{N}$ and
$t\in[i-\frac{1}{2},i+\frac{1}{2})$ we define:

$\mathcal{T}(\gamma(At)):=w_{i}.$

We need to show that this is a sublinear bi-Lipschitz equivalence. For any
given $t,t^{\prime}$, assume that $t\in[i-\frac{1}{2},i+\frac{1}{2})$ and
$t^{\prime}\in[j-\frac{1}{2},j+\frac{1}{2})$ and also assume without loss of
generality that $j-i$, we have that

$\displaystyle d(\mathcal{T}(\gamma(At),\gamma(At^{\prime}))$
$\displaystyle=d(w_{i},w_{j})\leq
d(w_{i},\gamma(Ai))+|j-i|A+d(w_{j},\gamma(Aj))$ $\displaystyle\leq
A|j-i|+2C\log\|w_{j}\|$ $\displaystyle\leq A|t-t^{\prime}+1|+2C\log\|w_{j}\|$
$\displaystyle\leq A|t-t^{\prime}|+2C\|\log w_{j}\|+1$

On the other hand we have

$\displaystyle d(\mathcal{T}(\gamma(At),\gamma(At^{\prime}))$
$\displaystyle=d(w_{i},w_{j})$ $\displaystyle\geq
A|j-i|-d(w_{i},\gamma(Ai))-d(w_{i},\gamma(Ai))$ $\displaystyle\geq
A|j-i|-2C\log j$ $\displaystyle\geq A|t-t^{\prime}|-A-2C\log\|w_{j}\|.$

Thus $\mathcal{T}$ is an SBE and the image of the geodesic $\gamma_{\omega}$
contains the sample path and is an $(A,\log n)$-ray. The proof for $G$ is a
relatively hylic group is identical, using also the facts that relatively
hylic group is weakly hyperbolic and there is also a distance formula that is
similar to that of the mapping class group [Sis13]. ∎

More generally, aside from the aforementioned two groups, the conclusion can
be applied to a wider range of countable groups with a compact boundary. We
say a boundary is _stably visible_ if any sequence of geodesics whose
endpoints converges to two distinct points on the boundary intersects some
bounded set of $X$. We can then combine the same argument as Theorem 5.1 and
apply Theorem 6 in [Tio15] to obtain the following:

######  Theorem 5.2.

Let $G$ be a countable group acting via isometries on a proper, geodesic,
metric space $(X,d)$ with a non-trivial, stably visible compactification. Let
$\mu$ be a probability measure on G with finite first moment with respect to
$d$, such that the semigroup generated by the support of $\mu$ is a non-
amenable group. Then there exists a constant $A$ and a sublinear function
$\kappa$ such that almost every sample path $(w_{n})$ is such that
$(w_{n}{\mathfrak{o}})$ is a $(A,\kappa)$-biLipschitz ray.

Example of such compactifications includes but are not limited to:

  1. (1)

the hyperbolic compactification of Gromov hyperbolic spaces;

  2. (2)

the end compactification of Freudenthal and Hopf [Hop44];

  3. (3)

the Floyd compactification (Section 3.2, [Tio15]);

  4. (4)

the visual compactification of a large class of CAT(0) spaces (Section 3.4,
[Tio15]).

  5. (5)

the redirecting compactification of asymptotically tree-graded spaces
([QR23]).

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