A finitely generated, locally indicable group with no faithful action by C^1 diffeomorphisms of the interval

According to Thurston's stability theorem, every group of C^1 diffeomorphisms of the closed interval is locally indicable (.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the semi-direct product between F_2 an Z^2, although locally indicable, does not embed into Diff_+^1 (]0,1[). (Here F_2 is any free subgroup of SL(2,Z), and its action on Z^2 is the projective one.) Moreover, we show that for every non-solvable subgroup G of SL(2,Z), the semi-direct product between G and Z^2 does not embed into Diff^1_+(S^1).Comment: 8 pages, no figur

Group actions on 1-manifolds: a list of very concrete open questions

This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure

What toleration means to contemporary liberalism?

Toleration is a key concept of liberalism, both from the historical and conceptual points of view. On the other hand, as people’s freedom to live according to their moral and religious ideas has long become a basic value for liberal societies and their political constitutions, it is reasonable to understand that there is nothing to be tolerated nor by citizens neither by the State. However, a part of the scope and meaning of the fundamental rights and freedoms is subject to what John Rawls calls reasonable disagreements and this is a field where toleration understood in the classic way is compatible with equality: not to intervene against that which is being disapproved understood has a raison d'être. Since the 1980s, toleration has been present in the debates on how to deal with pluralism in a constitutional democracy. This has to be connected to the rise of identity politics: political and intellectual movements such as multiculturalism or comunitarism that questioned whether social order based on neutral criteria was either possible or desirable or both things at the same time. Outstanding liberal philosophers were among those demanding political priority for comunitarian values and those who showed interest for toleration as a key concept to articulate pluralism. Key distinctions between them can be explained as the result of the different approaches they take when facing classical theories on toleration: whereas John Locke’s is a major influence on Rawls, John Stuart Mill’s is on the others, while Gray, Walzer and Rorty follow Isaiah Berlin’s reading of Mill.Universidad de Málaga, Campus de Excelencia Internacional Andalucía Tech