Asymptotic density of test elements in free groups and surface groups

2015/09/08 by Snopce, Ilir, Tanushevski, Slobodan · 1 citation
#20E05 #20E18 #20E36 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1509.02435

Abstract

An element g of a group G is a test element if every endomorphism of G that fixes g is an automorphism. Let G be a free group of finite rank, an orientable surface group of genus n ≥ 2, or a non-orientable surface group of genus n ≥ 3. Let T be the set of test elements of G. We prove that T is a net. From this result we derive that T has positive asymptotic density in G. This answers a question of Kapovich, Rivin, Schupp, and Shpilrain. Furthermore, we prove that T is dense in the profinite topology on G.

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• Algebraic groups over free and hyperbolic groups

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