| Title:
|
A note on normal generation and generation of groups (English) |
| Author:
|
Thom, Andreas |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
23 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
1-11 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\dots ,g_k$ with order $n_1,\dots ,n_k \in \{1,2,\dots \} \cup \{\infty \}$, then $$\beta _1^{(2)}(G) \leq k-1-\sum _{i=1}^{k} \frac 1{n_i}\,,$$ where $\beta _1^{(2)}(G)$ denotes the first $\ell ^2$-Betti number of $G$. We also show that any $k$-generated group with $\beta _1^{(2)}(G) \geq k-1-\varepsilon $ must have girth greater than or equal $1/\varepsilon $. (English) |
| Keyword:
|
group rings |
| Keyword:
|
$\ell ^2$-invariants |
| Keyword:
|
residually $p$-finite groups |
| Keyword:
|
normal generation |
| MSC:
|
16S34 |
| MSC:
|
46L10 |
| MSC:
|
46L50 |
| idZBL:
|
Zbl 1362.20026 |
| idMR:
|
MR3394074 |
| . |
| Date available:
|
2015-08-25T13:54:26Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144353 |
| . |
| Reference:
|
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| Reference:
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