Connectedness of spheres in Cayley graphs
We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or o...
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| Видавець: | Інститут прикладної математики і механіки НАН України |
|---|---|
| Дата: | 2018 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2018
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/188410 |
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| Цитувати: | Connectedness of spheres in Cayley graphs / J. Brieussel, A. Gournay // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 190–246. — Бібліогр.: 40 назв. — англ. |
Репозиторії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-188410 |
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irk-123456789-1884102023-02-28T01:27:04Z Connectedness of spheres in Cayley graphs Brieussel, J. Gournay, A. We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order n² to the sphere of radius n. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs. 2018 Article Connectedness of spheres in Cayley graphs / J. Brieussel, A. Gournay // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 190–246. — Бібліогр.: 40 назв. — англ. 1726-3255 2010 MSC: Primary 20F65; Secondary 20E22, 20F10. http://dspace.nbuv.gov.ua/handle/123456789/188410 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order n² to the sphere of radius n. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs. |
| format |
Article |
| author |
Brieussel, J. Gournay, A. |
| spellingShingle |
Brieussel, J. Gournay, A. Connectedness of spheres in Cayley graphs Algebra and Discrete Mathematics |
| author_facet |
Brieussel, J. Gournay, A. |
| author_sort |
Brieussel, J. |
| title |
Connectedness of spheres in Cayley graphs |
| title_short |
Connectedness of spheres in Cayley graphs |
| title_full |
Connectedness of spheres in Cayley graphs |
| title_fullStr |
Connectedness of spheres in Cayley graphs |
| title_full_unstemmed |
Connectedness of spheres in Cayley graphs |
| title_sort |
connectedness of spheres in cayley graphs |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188410 |
| citation_txt |
Connectedness of spheres in Cayley graphs / J. Brieussel, A. Gournay // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 190–246. — Бібліогр.: 40 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT brieusselj connectednessofspheresincayleygraphs AT gournaya connectednessofspheresincayleygraphs |
| first_indexed |
2023-10-18T23:08:18Z |
| last_indexed |
2023-10-18T23:08:18Z |
| _version_ |
1796157345935392768 |