Stability of amalgamated free products and HNN extensions

We study stability of amalgamated free products and HNN extensions of stable groups over finite groups. We focus on operator norm stability, Hilbert-Schmidt stability and stability in permutations. We provide many new examples of stable (or flexibly stable) non-amenable groups.Comment: Minor change

The Soil Classification course in Russian universities: an important ingredient of education

Teaching soil classification in the universities of Russia is discussed as a rather new experience in the education of specialists in environmental science. The lecture courses (24-30 academic hours) change following the inevitable changes in soil classification systems. As introduction, an outline of objectives and structure of soil classifications is given followed by a brief overview of most known national systems, which is also important for understanding the difficulties, origin and problems of the International WRB system. The latter is the central point of the course, its principles are explained, main diagnostic features of RSGs are communicated, and students are trained to use it basing on the descriptions of soil profiles and analytical data to them. In conclusion, the reasons to know and to work with soil classifications are specified as for future soil scientists, so for specialists in environmental science.

Local Hilbert--Schmidt stability

We introduce a notion of local Hilbert--Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert--Schmidt stable but not Hilbert--Schmidt stable. For amenable groups, we provide a criterion for local Hilbert--Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert--Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert--Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert--Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.Comment: 29 pages, comments welcom

Fire retainment on Cayley graphs

We study the fire-retaining problem on groups, a quasi-isometry invariant introduced by Mart\'inez-Pedroza and Prytula [8], related to the firefighter problem. We prove that any Cayley graph with degree-$d$ polynomial growth does not satisfy $\{f(n)\}$-retainment, for any $f(n) = o(n^{d-2})$, matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any $d\geq 1$, groups that satisfy $\{n^{d}\}$-retainment but not $o(n^d)$-retainment, as well as groups that do not satisfy sub-exponential retainment.Comment: 15 pages, 1 figur

Probabilistic Laws on Infinite Groups

We study the probability that certain laws are satisfied on infinite groups, focusing on elements sampled by random walks. For several group laws, including the metabelian one, we construct examples of infinite groups for which the law holds with high probability, but the group does not satisfy the law virtually. On the other hand, we show that if an infinite group satisfies the law $x^2=1$ with positive probability, then it is virtually abelian.Comment: 39 pages,1 figur

Unitary Lp+L^{p+}-representations of almost automorphism groups

Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze-Stein property, and let $\pi$ be a unitary representation of $H$. We show that the representation $\widetilde{\pi}$ of $G$ induced from $\pi$ is an $L^{p+}$-representation if and only if $\pi$ is an $L^{p+}$-representation. We deduce the following consequence for a large natural class of almost automorphism groups $G$ of trees: For every $p \in (2,\infty)$, the group $G$ has a unitary $L^{p+}$-representation that is not an $L^{q+}$-representation for any $q < p$. This in particular applies to the (coloured) Neretin groups.Comment: 4 page

The Impact of White Pine (Pinus strobus) on a Mollisol After Seven Decades of Soil Development

Selected chemical, physical, and macro and micromorphological properties in two pedons of a Clarion soil (Fine-loamy, mixed, superactive, mesic Typic Hapludolls) formed in till parent material, one planted to white pines (Pinus strobus) for the past 75 years and the other to grass, were compared. The most obvious difference between the two was the increased biological activity under pines; the variety and quantity of excrements suggested the activity of soil microfauna and variability of species resulted in finer and better aggregation of biological origin (crumbs and granules), numerous excrements in voids, and higher total porosity under pines. The matrix was lighter colored in the upper horizons under pines. The soil under pines seemed to be drier and to have more expressed water oscillations in the middle part of profile. There was some evidence of higher groundmass activity in the soil under pines and the groundmass b-fabric was slightly better expressed. The soil under the pines exhibited evidence of stronger weathering (weathered biotite at a shallower depth, more iron-rich fine fraction, common amorphous iron impregnation and frequent amorphous iron coatings related to grains or pores together with abundant iron nodules) than the soil under grass. Analytical and micromorphology methods showed only slight changes in the Clarion soil under pines. That means 75 years, at least under the prevailing climate, is too short a period for the formation of pronounced morphological and physico-chemical differences

Naucnoe tvorcestvo professora Marii Al`fredovny Glazovskoj

Przed-stawiono analizę drogi życiowej oraz twórczości naukowej znakomitego geografa, gleboznawcy i geochemika – M. A. Głazowskiej (1912–2016). W ciągu ¾ wieku badań opublikowała ponad 20 podręczników i monografii, a na tych dzie-łach oraz na Jej wykładach wychowało się wiele pokoleń geografów. M. A. Głazowska stworzyła szkołę naukową geo-chemii krajobrazów i geografii gleb