Geodesic
A note on normal generation and generation of groups
Communications in Mathematics, Tome 23 (2015) no. 1, pp. 1-11.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 $.
Classification : 16S34, 46L10, 46L50
Keywords: group rings; $\ell ^2$-invariants; residually $p$-finite groups; normal generation 
@article{COMIM_2015__23_1_a0,
     author = {Thom, Andreas},
     title = {A note on normal generation and generation of groups},
     journal = {Communications in Mathematics},
     pages = {1--11},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2015},
     mrnumber = {3394074},
     zbl = {1362.20026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2015__23_1_a0/}
}
TY  - JOUR
AU  - Thom, Andreas
TI  - A note on normal generation and generation of groups
JO  - Communications in Mathematics
PY  - 2015
SP  - 1
EP  - 11
VL  - 23
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/COMIM_2015__23_1_a0/
LA  - en
ID  - COMIM_2015__23_1_a0
ER  - 
%0 Journal Article
%A Thom, Andreas
%T A note on normal generation and generation of groups
%J Communications in Mathematics
%D 2015
%P 1-11
%V 23
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/COMIM_2015__23_1_a0/
%G en
%F COMIM_2015__23_1_a0
Thom, Andreas. A note on normal generation and generation of groups. Communications in Mathematics, Tome 23 (2015) no. 1, pp. 1-11. http://geodesic.mathdoc.fr/item/COMIM_2015__23_1_a0/