A note on normal generation and generation of groups

Andreas Thom

Communications in Mathematics (2015)

  • Volume: 23, Issue: 1, page 1-11
  • ISSN: 1804-1388

Abstract

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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 , ⋯ , g k with order n 1 , ⋯ , n k ∈ { 1 , 2 , ⋯ } ∪ { ∞ } , then β 1 ( 2 ) ( G ) ≤ k - 1 - ∑ i = 1 k 1 n i , where β 1 ( 2 ) ( G ) denotes the first ℓ 2 -Betti number of G . We also show that any k -generated group with β 1 ( 2 ) ( G ) ≥ k - 1 - ε must have girth greater than or equal 1 / ε .

How to cite

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Thom, Andreas. "A note on normal generation and generation of groups." Communications in Mathematics 23.1 (2015): 1-11. <http://eudml.org/doc/271593>.

@article{Thom2015,
abstract = {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 \lbrace 1,2,\dots \rbrace \cup \lbrace \infty \rbrace $, then \[\beta \_1^\{(2)\}(G) \le k-1-\sum \_\{i=1\}^\{k\} \frac\{1\}\{n\_i\}\,,\] 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) \ge k-1-\varepsilon $ must have girth greater than or equal $1/\varepsilon $.},
author = {Thom, Andreas},
journal = {Communications in Mathematics},
keywords = {group rings; $\ell ^2$-invariants; residually $p$-finite groups; normal generation},
language = {eng},
number = {1},
pages = {1-11},
publisher = {University of Ostrava},
title = {A note on normal generation and generation of groups},
url = {http://eudml.org/doc/271593},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Thom, Andreas
TI - A note on normal generation and generation of groups
JO - Communications in Mathematics
PY - 2015
PB - University of Ostrava
VL - 23
IS - 1
SP - 1
EP - 11
AB - 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 \lbrace 1,2,\dots \rbrace \cup \lbrace \infty \rbrace $, then \[\beta _1^{(2)}(G) \le k-1-\sum _{i=1}^{k} \frac{1}{n_i}\,,\] 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) \ge k-1-\varepsilon $ must have girth greater than or equal $1/\varepsilon $.
LA - eng
KW - group rings; $\ell ^2$-invariants; residually $p$-finite groups; normal generation
UR - http://eudml.org/doc/271593
ER -

References

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